pre-preparation course exam 2004-08.pdf

iranNational Math Olympiad (3rd Round)

1997

2nd round

1 Let a, b, c be real numbers. Prove that there exists a triangle with side lengths a, b, c if andonly if 2(a4 + b4 + c4) > (a2 + b2 + c2)2.

2 Prove that if a, b, c, dare positive integers such that ad = bc, then a + b + c + dcannot be aprime number.

3 Let N be the midpoint of side BC of triangle ABC. Right isosceles triangles ABM and ACPare constructed outside the triangle, with bases AB and AC. Prove that 4MNP is also aright isosceles triangle.

4 Let n blue points Ai and n red points Bi (i = 1, 2, ..., n) be situated on a line. Prove that∑i,j AiBj ≥

∑i<j AiAj +

∑i<j BiBj

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1997

3rd round

1 Suppose that S is a finite set of real numbers with the property that any two distinct elementsof S form an arithmetic progression with another element inS. Give an example of such a setwith 5 elements and show that no such set exists with more than 5 elements.

2 Suppose that 10 points are given in the plane, such that among any five of them there arefour lying on a circle. Find the minimum number of these points which must lie on a circle.

3 Consider a circle with diameter AB and center O, and let C and D be two points on thiscircle. The line CD meets the line AB at a point M satisfying MB < MA and MD < MC.Let K be the point of intersection (different from O) of the circumcircles of triangles AOCand DOB. Show that the lines MK and OK are perpendicular to each other, i. e. that]MKO = 90.

4 Determine all functions f : N0 → N0 − 1 such that for all n > 0 f(n + 1) + f(n + 3) =f(n + 5)f(n + 7)− 1375.

5 Let O be the circumcenter and H the orthocenter of an acute-angled triangle ABC such thatBC > CA. Let F be the foot of the altitude CH of triangle ABC. The perpendicular to theline OF at the point F intersects the line AC at P . Prove that ]FHP = ]BAC.

6 LetA be a symmetric 0, 1-matrix with all the diagonal entries equal to 1. Show that thereexist indices i1 < i2 < · · · < ik ≤ nsuch that Ai1 + Ai2 + ... + Aik = (1, 1, ..., 1) ( mod 2)where Ai denotes the i− th column of A.













1997

4th round

1 Let n be a positive integer. Prove that there exist polynomialsf(x)and g(x) with integercoefficients such that f(x)(x + 1)2

n+ g(x)(x2n

+ 1) = 2.

2 Suppose that f : R → R has the following properties: (i) f(x) = 1 for all x; (ii) f(x+ 1342)+

f(x) = f(x + 16) + f(x + 1

7) for all x. Prove that f is periodic.

3 Letω1, ω2, ..., ωkbe distinct real numbers with a nonzero sum. Prove that there exist integersn1, n2, ..., nk such that

∑ki=1 niωi > 0and for any non-identical permutationπ of 1, 2, . . . , k,

∑ki=1 niωπ(i) <

0

4 Let P be a variable point on arcBC of the circumcircle of triangle ABC not containing A.Let I1 and I2 be the incenters of the triangles PAB and PAC respectively. Prove that: (a)The circumcircle of ?PI1I2 passes through a fixed point. (b) The circle with diameter I1I2

passes through a fixed point. (c) The midpoint of I1I2 lies on a fixed circle.

5 Suppose that f : R+ → R+ is a function such that for all x, y > 0 f(x+y)+f(f(x)+f(y)) =f(f(x + f(y)) + f(y + f(x))). Prove that f(x) = f−1(x).

6 A building consists of finitely many rooms which have been separated by walls. There aresome doors on some of these walls which can be used to go around the building. Assume itis possible to reach any room from any other room. Two fixed rooms are marked by S andE. A person starts walking from S and wants to reachE.

A Program P = (Pi)i∈I is anR,L-sequence. The person uses it as follows: After passingthrough the n-th door, he chooses the door just to the right or left from the door just passed,meaning thatPn is R orL, and gets through it. In a room with one door, any symbol meansselecting the door he has just passed. The person stops as soon as he reaches E. Prove thatthere is a (possibly infinite) program P with the property that, no matter how the structureof the building is, the person can reach E by following it.













1998

2nd round

1 Let x and y be positive integers such that3x2 + x = 4y2 + y. Prove that x− yis a square.

2 Let KL and KN be tangent to the circle C (withL,N on C), and letM be a point on theextension of KN beyond N . The circumcircle of triangle KLM meetsC again at P . PointQisthe foot of the perpendicular from N toML. Prove that MPQ = 2∠KML.

3 An n × ntable is filled with numbers −1, 0, 1 in such a manner that every row and columncontains exactly one 1 and one −1. Prove that the rows and columns can be reordered sothat in the resulting table each number has been replaced with its negative.

4 Let x1, x2, x3, x4 be positive numbers with the product 1. Prove that:∑4

i=1 x3i ≥ max

∑4i=1 xi,

∑4i=1

1x

5 In an acute triangle ABC,D is the foot of the altitude from A. The bisectors of the innerangles B and C respectively meet AD atE and F . If BE = CF , prove that ABC is anisosceles triangle.

6 Supposea, bare natural numbers such that: p = b4

√2a−b2a+b is a prime number. What is the

maximum possible value ofp?

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1998

3rd round

1 Suppose thata, b, x are positive integers such that: xa+b = abb prove that a = xand b = xx.

2 In an acute triangle ABC, points D,E, F are the feet of the altitudes from A,B, C, respec-tively. A line through D parallel to EF meetsAC at Q and AB at R. Lines BCand EFintersect atP . Prove that the circumcircle of triangle PQR passes through the midpoint ofBC.

3 LetS = x0, x1, . . . , xn be a finite set of numbers in the interval [0, 1] withx0 = 0 and x1 = 1.We consider pairwise distances between numbers in S. If every distance that appears, exceptthe distance 1, occurs at least twice, prove that all the xi are rational.

4 Let ABC and XY Z be two triangles. Define A1 = BC∩ZX, A2 = BC∩XY , B1 = CA∩XY ,B2 = CA ∩ Y Z, C1 = AB ∩ Y Z, C2 = AB ∩ ZX. Hereby, the abbreviation g ∩ h means thepoint of intersection of two lines g and h.

Prove that C1C2AB = A1A2

BC = B1B2CA holds if and only if A1C2

XZ = C1B2ZY = B1A2

Y X .

5 Let x, y, z be real numbers greater than 1 such that 1x + 1

y + 1z = 2 Prove that:

√x− 1+

√y − 1+√

z − 1 ≤√

x + y + z

6 LetP be the set of all points in Rn with rational coordinates. For points A,B ∈ P , one canmove from A to Bif the distance AB is 1. Prove that every point inP can be reached fromany other point in P by a finite sequence of moves if and only ifn ≥ 5.













1998

4th round

1 Let f1, f2, f3 : R → R be functions such that a1f1 + a2f2 + a3f3 is monotonous for alla1, a2, a3 ∈ R. Show that there exist real numbers c1, c2, c3, not all zero, such that c1f1(x) +c2f2(x) + c3f3(x) = 0 for all real x.

2 Let X be a set with n elements, and let A1, A2, ..., Am be subsets of X such that:

1) |Ai| = 3 for every i ∈ 1, 2, ...,m; 2) |Ai ∩ Aj | ≤ 1 for all i, j ∈ 1, 2, ...,m such thati 6= j.

Prove that there exists a subset A of X such that A has at least√

2n elements, and for everyi ∈ 1, 2, ...,m, the set A does not contain Ai.

3 The edges of a regular 2n − gon are colored red and blue in some fashion. A step consistsin recoloring each edge whose neighbors are both the same color in red, and recoloring eachedge whose neighbors are of opposite colors in blue. Prove that after 2n − 1 steps all of theedges will be red, and show that this neednt hold after fewer steps.

4 Let n1 < n2 < . . . be a sequence of natural numbers such that for i < j the decimal rep-resentation of ni does not occur as the leftmost digits of the decimal representation ofnj

. (For example, 137and 13729 cannot both occur in the sequence.) Prove that:∑

i1ni≤

1+ 12 + 1

3 + 14 + 1

5 + 16 + 1

7 + · · ·+ 19 ¡!– s:D –¿¡img src=”SMILIESP ATH/iconmrgreen.gif”alt =

” : D”title = ”Mr.Green”/ ><! − −s : D − − ><! − −s : D − − >< imgsrc =”SMILIESP ATH/iconmrgreen.gif”alt = ” : D”title = ”Mr.Green”/ ><!−−s : D−− >LetABCbeatriangleandDbethepointontheextensionofsideBCpastCsuchthatCD = AC.ThecircumcircleofACDintersectsthecirclewithdiameterBCagainatP.LetBPmeetACatEandCPmeetABatF.P rovethatthepointsD,E,Farecollinear.

56 Let K be a convex polygon in the plane. Show that for any triangle of the minimum possiblearea containing K, the midpoints of its sides lie on K













1999

1 Define the sequence (xn) by x0 = 1 and for all n ∈ N,

xn =

xn−1 + (3r − 1)/2, if n = 3r−1(3k + 1);xn−1 − (3r + 1)/2, if n = 3r−1(3k + 2).

where k ∈ N0, r ∈ N. Prove that every integer occurs in this sequence exactly once.

2 Let n(r) be the maximum possible number of points with integer coordinates on a circle withradius r in Cartesian plane. Prove that n(r) < 6 3

√3πr2.

3 Let ABCDEF be a convex hexagon such that AB = BC, CD = DE and EF = FA. Provethat

AB

AD+

CD

CF+

EF

EB≥ 3

2.

4 Find all functions f : R→ R such that for all x, y,

f(f(x) + y) = f(x2 − y) + 4f(x)y.

5 In a triangle ABC, the bisector of angle BAC intersects BC at D. The circle Γ through Awhich is tangent to BC at D meets AC again at M . Line BM meets Γ again at P . Provethat line AP is a median of 4ABD.

6 Let ABC be a given triangle. Consider any painting of points of the plane in red and green.Show that there exist either two red points on the distance 1, or three green points forminga triangle congruent to 4ABC.













2001

1 Find all functions f : Q −→ Q such that: f(x) + f( 1x) = 1 2f(f(x)) = f(2x)

2 Does there exist a sequence bi∞i=1 of positive real numbers such that for each natural m:

bm + b2m + b3m + · · · = 1m









2002

1 Let a, b, c ∈ Rn, a + b + c = 0 and λ > 0. Prove that∏cycle

|a|+ |b|+ (2λ + 1)|c||a|+ |b|+ |c|

≥ (2λ + 3)3

2 f : R −→ R+ is a non-decreasing function. Prove that there is a point a ∈ R that

f(a +1

f(a)) < 2f(a)

3 an is a sequence that a1 = 1, a2 = 2, a3 = 3, and

an+1 = an − an−1 +a2

n

an−2

Prove that for each natural n, an is integer.

4 an (n is integer) is a sequence from positive reals that

an ≥an+2 + an+1 + an−1 + an−2

4

Prove an is constant.

5 ω is circumcirlce of triangle ABC. We draw a line parallel to BC that intersects AB,AC atE,F and intersects ω at U, V . Assume that M is midpoint of BC. Let ω′ be circumcircle ofUMV . We know that R(ABC) = R(UMV ). ME and ω′ intersect at T , and FT intersectsω′ at S. Prove that EF is tangent to circumcircle of MCS.

6 M is midpoint of BC.P is an arbitary point on BC. C1 is tangent to big circle.Suppose radius of C1

is r1 Radius of C4 is equal to radius of C1 and C4 is tangent to BC at P. C2 and C3 are tangent to bigcircle and line BC and circle C4. [img]http://aycu01.webshots.com/image/4120/2005120338156776027rs.jpg[/img]Prove :r1 + r2 + r3 = R(R radius of big circle)

In triangle ABC, AD is angle bisector (D is on BC) if AB + AD = CD and AC + AD = BC,what are the angles of ABC?

Circles C1 and C2 are tangent to each other at K and are tangent to circle C at M and N . Externaltangent of C1 and C2 intersect C at A and B. AK and BK intersect with circle C at E and Frespectively. If AB is diameter of C, prove that EF and MN and OK are concurrent. (O is centerof circle C.)

Let M and N be points on the side BC of triangle ABC, with the point M lying on the segmentBN , such that BM = CN . Let P and Q be points on the segments AN and AM , respectively,such that ]PMC = ]MAB and ]QNB = ]NAC. Prove that ]QBC = ]PCB.













2002

H, I, O, N are orthogonal center, incenter, circumcenter, and Nagelian point of triangle ABC.Ia, Ib, Ic are excenters of ABC corresponding vertices A,B, C. S is point that O is midpoint ofHS. Prove that centroid of triangles IaIbIc and SIN concide.

In an m × n table there is a policeman in cell (1, 1), and there is a thief in cell (i, j). A move isgoing from a cell to a neighbor (each cell has at most four neighbors). Thief makes the first move,then the policeman moves and ... For which (i, j) the policeman can catch the thief?

We have a bipartite graph G (with parts X and Y ). We orient each edge arbitrarily. Hessam choosesa vertex at each turn and reverse the orientation of all edges that v is one of their endpoint. Provethat with these steps we can reach to a graph that for each vertex v in part X, deg+(v) ≥ deg−(v)and for each vertex in part Y , deg+v ≤ deg−v

f, g are two permutations of set X = 1, . . . , n. We say f, g have common points iff there is ak ∈ X that f(k) = g(k). a) If m > n

2 , prove that there are m permutations f1, f2, . . . , fm from Xthat for each permutation f ∈ X, there is an index i that f, fi have common points. b) Prove thatif m ≤ n

2 , we can not find permutations f1, f2, . . . , fm satisfying the above condition.

A subset S of N is eventually linear iff there are k, N ∈ N that for n > N,n ∈ S ⇐⇒ k|n. Let Sbe a subset of N that is closed under addition. Prove that S is eventually linear.

Let A be be a point outside the circle C, and AB and AC be the two tangents from A to thiscircle C. Let L be an arbitrary tangent to C that cuts AB and AC in P and Q. A line throughP parallel to AC cuts BC in R. Prove that while L varies, QR passes through a fixed point.¡!– s:) –¿¡img src=”SMILIESP ATH/smile.gif”alt = ” :)”title = ”Smile”/ ><! − −s :) − − >Forpositivea,b,c, a2 + b2 + c2 + abc = 4Prove a + b + c ≤ 3

Find the smallest natural number n that the following statement holds : Let A be a finite subsetof R2. For each n points in A there are two lines including these n points. All of the points lie ontwo lines.

Find all continious f : R −→ R that for any x, y

f(x) + f(y) + f(xy) = f(x + y + xy)

I is incenter of triangle ABC. Incircle of ABC touches AB,AC at X, Y . XI intersects incircle atM . Let CM ∩ AB = X ′. L is a point on the segment X ′C that X ′L = CM . Prove that A,L, Iare collinear iff AB = AC.

a0 = 2, a1 = 1 and for n ≥ 1 we know that : an+1 = an + an−1 m is an even number and p is primenumber such that p divides am − 2. Prove that p divides am+1 − 1.

Excircle of triangle ABC corresponding vertex A, is tangent to BC at P . AP intersects circumcircleof ABC at D. Prove

r(PCD) = r(PBD)

whcih r(PCD) and r(PBD) are inradii of triangles PCD and PBD.







2002

15000 years ago Tilif ministry in Persia decided to define a code for n ≥ 2 cities. Each code isa sequence of 0, 1 such that no code start with another code. We know that from 2m calls fromforeign countries to Persia 2m−ai of them where from the i-th city (So

∑ni=1

12ai = 1). Let li be

length of code assigned to i-th city. Prove that∑n

i=1li2i is minimum iff ∀i, li = ai

Find all polynomials p with real coefficients that if for a real a,p(a) is integer then a is integer.

A,B, C are on circle C. I is incenter of ABC , D is midpoint of arc BAC. W is a circle that istangent to AB and AC and tangent to C at P . (W is in C) Prove that P and I and D are on aline.

An ant walks on the interior surface of a cube, he moves on a straight line. If ant reaches to an edgethe he moves on a straight line on cube’s net. Also if he reaches to a vertex he will return his path.a) Prove that for each beginning point ant can has infinitely many choices for his direction thatits path becomes periodic. b) Prove that if if the ant starts from point A and its path is periodic,then for each point B if ant starts with this direction, then his path becomes periodic.







2003

1 suppose this equation: x ¡sup¿2¡/sup¿ +y ¡sup¿2¡/sup¿ +z ¡sup¿2¡/sup¿ =w ¡sup¿2¡/sup¿ .show that the solution of this equation ( if w,z have same parity) are in this form: x=2d(XZ-YW), y=2d(XW+YZ),z=d(X ¡sup¿2¡/sup¿ +Y ¡sup¿2¡/sup¿ -Z ¡sup¿2¡/sup¿ -W ¡sup¿2¡/sup¿),w=d(X ¡sup¿2¡/sup¿ +Y ¡sup¿2¡/sup¿ +Z ¡sup¿2¡/sup¿ +W ¡sup¿2¡/sup¿ )

2 assume ABCD a convex quadrilatral. P and Q are on BC and DC respectively such thatangle BAP= angle DAQ .prove that [ADQ]=[ABP] ([ABC] means its area ) iff the line whichcrosses through the orthocenters of these traingles , is perpendicular to AC.

3 assume that A is finite subset of prime number and a is an positive integer prove that ther arefinite positive integer like m s.t: prime divisors of am−1arecontainedinA.XOY isangleintheplane.A,BarevariablepointonOX,OY suchthat1/OA+1/OB = 1/K(kisconstant).drawtwocircleswithdiameterOAandOB.provethatcommonexternaltangenttothesecirclesistangenttotheconstantcircle(diterminetheradiusandthelocusofitscenter).

45 assume P is a odd prime number and S is the sum of the all primitive root mod P. show that If p-1isn’t empty of perfect square( I mean that we can show p-1 =k ¡sup¿2¡/sup¿m) then S=0(mod P).if not S=?(mod p).

6 let the incircle of a triangle ABC touch BC,AC,AB at A1,B1,C1 respectively. M and N are themidpoints of AB1 and AC1 respectively. MN meets A1C1 at T . draw two tangents TP and TQthrough T to incircle. PQ meets MN at L and B1C1 meets PQ at K . assume I is the center of theincircle .

prove IK is parallel to AL

7 f1, f2, . . . , fn are polynomials with integer coefficients. Prove there exist a reducible g(x) withinteger coefficients that f1 + g, f2 + g, . . . , fn + g are irreducible.

8 Let’s call perfect power any positive integer n such that n = ab for some integers a and b, withb > 1. a) Find 2004 perfect powers in arithmetic progression. b) Prove that perfect powers cannotform an infinite arithmetic progression.

9 Does there exist an infinite set S such that for every a, b ∈ S we have a2 + b2 − ab | (ab)2.

10 let p be a prime and a and n be natural numbers such that (pa−1)/(p−1) = 2nfindthenumberofnaturaldivisorsofna. <!−−s :)−− >< imgsrc = ”SMILIESP ATH/smile.gif”alt = ” :)”title = ”Smile”/ ><!−−s :)−− > assumethatXisasetofnnumber.and0≤ k ≤ n.the maximum number of permutation whichacting on X st every two of them have at least k component in common,is an,k.and the maximumnuber of permutation st every two of them have at most k component in common,is bn,k. a)proevethat :an,k · bn,k−1 ≤ n! b)assume that p is prime number,determine the exact value of ap,2.

1112 There is a lamp in space.(Consider lamp a point) Do there exist finite number of equal sphers inspace that the light of the lamp can not go to the infinite?(If a ray crash in a sphere it stops)



















2003

13 here is the most difficult and the most beautiful problem occurs in 21th iranian (2003) olympiadassume that P is n-gon ,lying on the plane ,we name its edge 1,2,..,n. if S=s1,s2,s3,.... be a finiteor infinite sequence such that for each i, si is in 1,2,...,n, we move P on the plane according tothe S in this form: at first we reflect P through the s1 ( s1 means the edge which iys number iss1)then through s2 and so on like the figure below. a)show that there exist the infinite sequence Ssucth that if we move P according to S we cover all the plane b)prove that the sequence in a) isn’tperiodic. c)assume that P is regular pentagon ,which the radius of its circumcircle is 1,and D iscircle ,with radius 1.00001 ,arbitrarily in the plane .does exist a sequence S such that we move Paccording to S then P reside in D completely?

14 n≥ 6isaninteger.evaluatetheminimumoff(n)s.t : anygraphwithnverticesandf(n)edgecontainstwocyclewhicharedistinct(alsotheyhavenocomonvertice)?Assumem×nmatrix which is filled with just 0, 1 and any two row differ in at least n/2 members, show thatm ≤ 2n. ( for example the diffrence of this two row is only in one index 110 100)

Edited by Myth

1516 Segment AB is fixed in plane. Find the largest n, such that there are n points P1, P2, . . . , Pn inplane that triangles ABPi are similar for 1 ≤ i ≤ n. Prove that all of Pi’s lie on a circle.

17 A simple calculator is given to you. (It contains 8 digits and only does the operations +,-,*,/,√)

How can you find 3√

2 with accuracy of 6 digits.

18 In tetrahedron ABCD, radius four circumcircles of four faces are equal. Prove that AB = CD,AC = BD and AD = BC.

19 An integer n is called a good number if and only if |n| is not square of another intger. Find allintegers m such that they can be written in infinitely many ways as sum of three different goodnumbers and product of these three numbers is square of an odd number.

20 Suppose that M is an arbitrary point on side BC of triangle ABC. B1, C1 are points on AB,ACsuch that MB = MB1 and MC = MC1. Suppose that H, I are orthocenter of triangle ABC andincenter of triangle MB1C1. Prove that A,B1,H, I, C1 lie on a circle.

21 Let ABC be a triangle. Wa is a circle with center on BC passing through A and perpendicular tocircumcircle of ABC. Wb,Wc are defined similarly. Prove that center of Wa,Wb,Wc are collinear.

22 Let a1 = a2 = 1 and

an+2 =n(n + 1)an+1 + n2an + 5

n + 2− 2

for each n ∈ N. Find all n such that an ∈ N.

23 Find all homogeneous linear recursive sequences such that there is a T such that an = an+T foreach n.


















2003

24 A,B are fixed points. Variable line l passes through the fixed point C. There are two circles passingthrough A,B and tangent to l at M,N . Prove that circumcircle of AMN passes through a fixedpoint.

25 Let A,B, C, Q be fixed points on plane. M,N,P are intersection points of AQ,BQ, CQ withBC, CA, AB. D′, E′, F ′ are tangency points of incircle of ABC with BC, CA, AB. Tangentsdrawn from M,N,P (not triangle sides) to incircle of ABC make triangle DEF . Prove thatDD′, EE′, FF ′ intersect at Q.

26 Circles C1, C2 intersect at P . A line ∆ is drawn arbitrarily from P and intersects with C1, C2 atB,C. What is locus of A such that the median of AM of triangle ABC has fixed length k.

27 S ⊂ N is called a square set, iff for each x, y ∈ S, xy + 1 is square of an integer. a) Is S finite? b)Find maximum number of elements of S.

28 There are n points in R3 such that every three form an acute angled triangle. Find maximum of n.

29 Let c ∈ C and Ac = p ∈ C[z]|p(z2 + c) = p(z)2 + c.a) Prove that for each c ∈ C, Ac is infinite.

b) Prove that if p ∈ A1, and p(z0) = 0, then |z0| < 1.7.

c) Prove that each element of Ac is odd or even.

Let fc = z2 + c ∈ C[z]. We see easily that Bc := z, fc(z), fc(fc(z)), . . . is a subset of Ac. Provethat in the following cases Ac = Bc.

d) |c| > 2.

e) c ∈ Q\Z.

f) c is a non-algebraic number

g) c is a real number and c 6∈ [−2, 14 ].













2004

1 We say m n for natural m,n ⇐⇒ nth number of binary representation of m is 1 or mthnumber of binary representation of n is 1. and we say m • n if and only if m,n doesn’t havethe relation We say A ⊂ N is golden ⇐⇒ ∀U, V ⊂ A that are finite and arenot empty andU ∩V = ∅,There exist z ∈ A that ∀x ∈ U, y ∈ V we have z x, z • y Suppose P is set of primenumbers.Prove if P = P1 ∪ ... ∪ Pk and Pi ∩ Pj = ∅ then one of P1, ..., Pk is golden.

2 A is a convex set in plane prove taht ther exist O in A that for every line XX ′ passing throwO and X and X ′ are boundry points of A then

12≤ OX

OX ′ ≤ 2

3 Suppose V = Zn2 and for a vector x = (x1, ..xn) in V and permutation σ.We have xσ =

(xσ(1), ..., xσ(n)) Suppose n = 4k + 2, 4k + 3 and f : V 7→ V is injective and if x and y differin more than n/2 places then f(x) and f(y) differ in more than n/2 places. Prove there existpermutaion σ and vector v that f(x) = xσ + v

4 We have finite white and finite black points that for each 4 oints there is a line that whitepoints and black points are at different sides of this line.Prove there is a line that all whitepoints and black points are at different side of this line.

5 assume that k,n are two positive integer k ≤ ncount the number of permutation 1, . . . , n stfor any 1 ≤ i, j ≤ kand any positive integer m we have fm(i) 6= j (fm meas iterarte function,)

6 assume that we have a n*n table we fill it with 1,...,n such that each number exists exactlyn times prove that there exist a row or column such that at least

√n diffrent number are

contained.

7 Suppose F is a polygon with lattice vertices and sides parralell to x-axis and y-axis.SupposeS(F ), P (F ) are area and perimeter of F . Find the smallest k that: S(F ) ≤ k.P (F )2

8 P is a n-gon with sides l1, ..., ln and vertices on a circle. Prove that no n-gon with this sideshas area more than P

9 Let ABC be a triangle, and O the center of its circumcircle. Let a line through the point Ointersect the lines AB and AC at the points M and N , respectively. Denote by S and R themidpoints of the segments BN and CM , respectively. Prove that ]ROS = ]BAC.

10 f : R2 7→ R2 is injective and surjective.Distance of X and Y is not less than distance of f(X)and f(Y ).Prove for A in plane:

S(A) ≥ S(f(A))

Which S(A) is area of A

















2004

11 assume that ABC is acute traingle and AA’ is median we extend it until it meets circumcircleat A”. let APa be a diameter of the circumcircle. the pependicular from A’ to APa meetsthe tangent to circumcircle at A” in the point Xa; we define Xb, Xc similary . prove thatXa, Xb, Xc are one a line.

12 N10 is generalization of N that every hypernumber in N10 is something like: ...a2a1a0 withai ∈ 0, 1..9 (Notice that ...000 ∈ N10) Also we easily have +, ∗ in N10. first k number of a∗ b=first k nubmer of (first k number of a * first k number of b) first k number of a + b= first knubmer of (first k number of a + first k number of b) Fore example ...999 + ...0001 = ...000Prove that every monic polynomial in N10[x] with degree d has at most d2 roots.

13 Suppose f is a polynomial in Z[X] and m is integer .Consider the sequence ai like this a1 = mand ai+1 = f(ai) find all polynomials f and alll integers m that for each i:

ai|ai+1

14 We define f : N → N, f(n) =∑n

k=1(k, n).

a) Show that if gcd(m,n) = 1 then we have f(mn) = f(m) · f(n);

b) Show that∑

d|n f(d) = nd(n).

15 This problem is easy but nobody solved it. point A moves in a line with speed v and Bmoves also with speed v′ that at every time the direction of move of B goes from A.We knowv ≥ v′.If we know the point of beginning of path of A, then B must be where at first that Bcan catch A.

16 Let ABC be a triangle . Let point X be in the triangle and AX intersects BC in Y . Draw theperpendiculars Y P, Y Q, Y R, Y S to lines CA, CX, BX,BA respectively. Find the necessaryand sufficient condition for X such that PQRS be cyclic .

17 Let p = 4k + 1 be a prime. Prove that p has at least φ(p−1)2 primitive roots.

18 Prove that for each n, there is a subset of N such as a1, . . . , an that for each subset S of1, . . . , n,

∑i∈S ai has the same set of prime divisors.

19 Find all integer solutions of p3 = p2 + q2 + r2 where p, q, r are primes.

20 p(x) is a polynomial in Z[x] such that for each m,n ∈ N there is an integer a such thatn|p(am). Prove that 0 or 1 is a root of p(x).

21 a1, a2, . . . , an are integers and∑

i,j(ai − aj)2 > 0. Prove that there are infinitely many primenumbers such as p that for some k:

p|ak1 + · · ·+ ak

n


















2004

22 Suppose that F is a family of subsets of X. A,B are two subsets of X s.t. each element ofX has non-empty intersection with A,B. We know that no subset of X with n− 1 elementshas this property. Prove that there is a representation A,B in the form A = a1, . . . , an andB = b1, . . . , bn such that for each 1 ≤ i ≤ n, there is an element of F containing both ai, bi.

23 F is a family of 3-subsets of set X. Every two distinct elements of X are exactly in k elementsof F . It is known that there is a partition of F to sets X1, X2 such that each element of Fhas non-empty intersection with both X1, X2. Prove that |X| ≤ 4.

24 In triangle ABC, points M,N lie on line AC such that MA = AB and NB = NC. AlsoK, L lie on line BC such that KA = KB and LA = LC. It is know that KL = 1

2BC andMN = AC. Find angles of triangle ABC.

25 Finitely many convex subsets of R3 is given such that every three have non-empty intersection.Prove that there is a line in R3 such that intersects all of these subsets.

26 Finitely many points are given on the surface of a sphere, such that every four of them lie onthe surface of open hemisphere. Prove that all points lie on the surface of an open hemisphere.

27 ∆1, . . . ,∆n are n concurrent segments (their lines concur) in the real plane. Prove that if forevery three of them there is a line intersecting these three segments, then there is a line thatinteresects all of segments.

28 Find all prime numbers such that p = m2 + n2 and p|m3 + n3 − 4.

29 Incircle of triangle ABC touches AB,AC at P,Q. BI,CI intersect with PQ at K, L. Provethat circumcircle of ILK is tangent to incircle of ABC if and only if AB + AC = 3BC.

30 Find all polynomials p ∈ Z[x] such that (m,n) = 1 ⇒ (p(m), p(n)) = 1
















2005

Day 1

1 Suppose a, b, c ∈ R+. Prove that :

(a

b+

b

c+

c

a)2 ≥ (a + b + c)(

1a

+1b

+1c)

2 Suppose xn is a decreasing sequence that limn→∞

xn = 0. Prove that∑

(−1)nxn is convergent

3 Find all α > 0 and β > 0 that for each (x1, . . . , xn) and (y1, . . . , yn) ∈ R+n that:

(∑

xαi )(

∑yβ

i ) ≥∑

xiyi

4 Suppose P,Q ∈ R[x] that deg P = deg Q and PQ′ − QP ′ has no real root. Prove that foreach λ ∈ R number of real roots of P and λP + (1− λ)Q are equal.

5 Suppose a, b, c ∈ R+and1

a2 + 1+

1b2 + 1

+1

c2 + 1= 2

Prove that ab + ac + bc ≤ 32

6 Suppose A ∈ Rm is closed and non-empty . f : A 7→ A is a lipchitz function with constantless than 1. (ie there exist c < 1 that |f(x)− f(y)| < |x− y|, ∀x, y ∈ A). Prove that f has aunique point like x that f(x) = x













2005

Day 2

1 From each vertex of triangle ABC we draw 3 arbitary parrallell lines, and from each vertexwe draw a perpendicular to these lines. There are 3 rectangles that one of their diagnals istriangle’s side. We draw their other diagnals and call them `1, `2 and `3.

a) Prove that `1, `2 and `3 are concurrent at a point P .

b) Find the locus of P as we move the 3 arbitary lines.

2 Suppose O is circumcenter of triangle ABC. Suppose S(OAB)+S(OAC)2 = S(OBC). Prove

that ditsance of O (circumcenter) from radial axis of circumcircle and 9-point circle is

a2√9R2 − (a2 + b2 + c2)

3 Prove that in acute-angled traingle ABC if r is inradius and R is radius of circumcircle then:

a2 + b2 + c2 ≥ 4(R + r)2

4 Suppose in triangle ABC incircle touches the side BC at P and ∠APB = α. Prove that :

1p− b

+1

p− c=

2rtgα

5 Suppose H and O are orthocenter and circumcenter of triangle ABC. ω is circumcircle ofABC. AO intersects with ω at A1. A1H intersects with ω at A′ and A′′ is the intersectionpoint of ω and AH. We define points B′, B′′, C ′ and C ′′ similiarly. Prove that A′A′′, B′B′′

and C ′C ′′ are concurrent in a point on the Euler line of triangle ABC.












2005

Day 3

1 Find all n, p, q ∈ N that:2n + n2 = 3p7q

2 a ∈ N and m = a2 + a + 1. Find the teh number of 0 ≤ x ≤ m that:

x3 ≡ 1(mod m)

3 p(x) is an irreducible polynomial in Q[x] that deg p is odd. q(x), r(x) are polynomials withrational coefficients that p(x)|q(x)2 + q(x).r(x) + r(x)2. Prove that

p(x)2|q(x)2 + q(x).r(x) + r(x)2

4 k is an integer. We define the sequence an∞n=0 like this:

a0 = 0, a1 = 1, an = 2kan−1 − (k2 + 1)an−2 (n ≥ 2)

p is a prime number that p ≡ 3(mod 4) a) Prove that an+p2−1 ≡ an(mod p) b) Prove thatan+p3−p ≡ an(mod p2)

5 a, b, c ∈ N that a, b 6= c. Prove that there are infinitely many prime numbers like P thatthere exist n ∈ N that p|an + bn − cn












2005

Day 4

1 We call the set A ∈ Rn CN if and only if every continuous f : A 7→ A ther exist : x ∈ A thatf(x) = x a)Example: We know that A = x ∈ Rn||x| ≤ 1 is CN b) circle is not CN. Whichone of thes sets are CN? 1) A = x ∈ R3||x| = 12) The cross (x, y) ∈ R2|xy = 0, |x|+ |y| ≤ 13) Graph of f : [0, 1] :7→ R:

f(x) = sin1x

if x 6= 0, f(0) = 0

2 n vectors are on the plane. We can move each vector forward and backeard on the line thatthe vector is on it. If there are 2 vectors that their endpoints concide we can omit them andreplace them with their sum (If their sum is nonzero). Suppose with these operations with 2different method we reach to a vector. Prove that these vectors are on a common line

3 f(n) is the least number that there exist a f(n)−mino that contains every n−mino. Provethat 10000 ≤ f(1384) ≤ 960000. Find some bound for f(n)

4 a) Year 1872 Texas 3 gold miners found a peice of gold. They have a coin that with possibilityof 1

2 it will come each side, and they want to give the piece of gold to one of themselvesdepending on how the coin will come. Design a fair method (It means that each of the 3miners will win the piece of gold with possibility of 1

3) for the miners.

b) Year 2005, faculty of Mathematics, Sharif university of Technolgy Suppose 0 < α < 1 andwe want to find a way for people name A and B that the possibity of winning of A is α. Isit possible to find this way?

c) Year 2005 Ahvaz, Takhti Stadium Two soccer teams have a contest. And we want tochoose each player’s side with the coin, But we don’t know that our coin is fair or not. Finda way to find that coin is fair or not?

d) Year 2005,summer In the National mathematical Oympiad in Iran. Each student has acoin and must find a way that the possibility of coin being TAIL is α or no. Find a way forthe student.











2005

Day 5

1 An airplane wants to go from a point on the equator, and at each moment it will go to thenortheast with speed v. Suppose the radius of earth is R. a) Will the airplane reach to thenorth pole? If yes how long it will take to reach the north pole? b) Will the airplne rotatefinitely many times around the north pole? If yes how many times?

2 We define a relation between subsets of Rn. A ∼ B ⇐⇒ we can partition A,B in sets

A1, . . . , An and B1, . . . , Bn(i.e A =n⋃

i=1

Ai, B =n⋃

i=1

Bi, Ai ∩ Aj = ∅, Bi ∩ Bj = ∅) and

Ai ' Bi. Say the the following sets have the relation ∼ or not ?

a) Natural numbers and composite numbers. b) Rational numbers and rational numbers withfinite digits in base 10. c) x ∈ Q|x <

√2 and x ∈ Q|x <

√3 d) A = (x, y) ∈ R2|x2+y2 <

1 and A \ (0, 0)

3 For each m ∈ N we define rad (m) =∏

pi that m =∏

pαii .

abc Conjecture Suppose ε > 0 is an arbitary number, then there exist K depinding on εthat for each 3 numbers a, b, c ∈ Z that gcd(a, b) = 1 and a + b = c then:

max|a|, |b|, |c| ≤ K(rad (abc))1+ε

Now prove each of the foollowing statements with abc conjectures : a) Fermat’s last theoremfor n > N that N is a natural number. b)We call n =

∏pαi

i strong if and only αi ≥ 2 c)Prove that there are finitely many n that n, n + 1, n + 2 are strong. d) Prove that there arefinitely many rational numbers like p

q that:

∣∣∣ 3√

2− p

q

∣∣∣ <21384

q3

4 Suppose we have some proteins that each protein is a sequence of 7 ”AMINO-ACIDS”A, B, C, H, F, N . For example AFHNNNHAFFC is a protein. There are some steps thatin each step an amino-acid will change to another one. For example with the step NA → Nthe protein BANANA will cahnge to BANNA(”in Persian means workman”). We have aset of allowed steps that each protein can change with these steps. For example with the setof steps:1) AA −→ A2) AB −→ BA3) A −→ null Protein ABBAABA will change like this:ABBAABA











2005

ABBABABABABABBAABABBABABBBAABBBABBB You see after finite steps this protein will finish it steps. Set of allowed steps thatfor them there exist a protein that may have infinitely many steps is dangerous. Which of

the following allowed sets are dangerous? a) NO −→ OONN b)

HHCC −→ HCCHCC −→ CH

c)

Design a set of allowed steps that change AA . . . A︸︷︷︸n

−→ BB . . . B︸︷︷︸2n

d) Design a set of allowed

steps that change A . . . A︸︷︷︸n

B . . . B︸︷︷︸m

−→ CC . . . C︸︷︷︸mn

You see from c and d that we acn calculate

the functions F (n) = 2n and G(M,N) = mn with these steps. Find some other calculatablefunctions with these steps. (It has some extra mark.)







2006

Algebra

1 For positive numbers x1, x2, . . . , xs, we know that∏s

i=1 xk = 1. Prove that for each m ≥ n

s∑k=1

xmk ≥

s∑k=1

xnk

2 Find all real polynomials that

p(x + p(x)) = p(x) + p(p(x))

3 Find all real x, y, z that x + y + zx = 1

2

y + z + xy = 12

z + x + yz = 12

4 p(x) is a real polynomial that for each x ≥ 0, p(x) ≥ 0. Prove that there are real polynomialsA(x), B(x) that p(x) = A(x)2 + xB(x)2

5 Find the biggest real number k such that for each right-angled triangle with sides a, b, c, wehave

a3 + b3 + c3 ≥ k (a + b + c)3 .

6 P,Q,R are non-zero polynomials that for each z ∈ C, P (z)Q(z) = R(z). a) If P,Q,R ∈ R[x]prove that Q is constant polynomial. b) IS the above statement correct for P,Q,R ∈ C[x]?













2006

Combinatorics

1 Let A be a family of subsets of 1, 2, . . . , n such that no member of A is contained in another.Sperners Theorem states that |A| ≤

(n

bn2c). Find all the families for which the equality holds.

2 Let B be a subset of Zn3 with the property that for every two distinct members (a1, . . . , an) and

(b1, . . . , bn) of B there exist 1 ≤ i ≤ n such that ai ≡ bi + 1 (mod 3). Prove that |B| ≤ 2n.

3 Let C be a (probably infinite) family of subsets of N such that for every chain C1 ⊂ C2 ⊂ . . .of members of C, there is a member of C containing all of them. Show that there is a memberof C such that no other member of C contains it!

4 Let D be a family of s-element subsets of 1. . . . , n such that every k members of D havenon-empty intersection. Denote by D(n, s, k) the maximum cardinality of such a family. a)Find D(n, s, 4). b) Find D(n, s, 3).

5 Let E be a family of subsets of 1, 2, . . . , n with the property that for each A ⊂ 1, 2, . . . , nthere exist B ∈ F such that n−d

2 ≤ |A4B| ≤ n+d2 . (where A4B = (A \B) ∪ (B \A) is the

symmetric difference). Denote by f(n, d) the minimum cardinality of such a family. a) Provethat if n is even then f(n, 0) ≤ n. b) Prove that if n − d is even then f(n, d) ≤ d n

d+1e. c)Prove that if n is even then f(n, 0) = n

6 The National Foundation of Happiness (NFoH) wants to estimate the happiness of people ofcountry. NFoH selected n random persons, and on every morning asked from each of themwhether she is happy or not. On any two distinct days, exactly half of the persons gave thesame answer. Show that after k days, there were at most n − n

k persons whose yes answersequals their no answers.













2006

Final Exam

1 A regular polyhedron is a polyhedron that is convex and all of its faces are regular polygons.We call a regular polhedron a ”Choombam” iff none of its faces are triangles. a) prove that eachchoombam can be inscribed in a sphere. b) Prove that faces of each choombam are polygons ofat most 3 kinds. (i.e. there is a set m,n, q that each face of a choombam is n-gon or m-gonor q-gon.) c) Prove that there is only one choombam that its faces are pentagon and hexagon.(Soccer ball) [img]http://aycu08.webshots.com/image/5367/2001362702285797426rs.jpg[/img]d)Forn¿3, aprismthatitsfacesare2regularn−gonsandnsquares, isachoombam.Provethatexceptthesechoombamstherearefinitelymanychoombams.Aliquidismovinginaninfinitepipe.Foreachmoleculeifitisatpointwithcoordinatexthenaftertsecondsitwillbeatapointofp(t,x).P rovethatifp(t,x)isapolynomialoft,xthenspeedofallmoleculesareequalandconstant.

23 For A ⊂ Z and a, b ∈ Z. We define aA + b := ax + b|x ∈ A. If a 6= 0 then we calll aA + b andA to similar sets. In this question the Cantor set C is the number of non-negative integers that intheir base-3 representation there is no 1 digit. You see

C = (3C)∪(3C + 2) (1)

(i.e. C is partitioned to sets 3C and 3C+2). We give another example C = (3C)∪(9C+6)∪(3C+2).

A representation of C is a partition of C to some similiar sets. i.e.

C =n⋃

i=1

Ci (2)

and Ci = aiC + bi are similar to C. We call a representation of C a primitive representation iffunion of some of Ci is not a set similar and not equal to C. Consider a primitive representationof Cantor set. Prove that a) ai > 1. b) ai are powers of 3. c) ai > bi d) (1) is the only primitiverepresentation of C.

4 The image shown below is a cross with length 2. If length of a cross of length k it is called a k-cross.(Each k-cross ahs 6k+1 squares.) [img]http://aycu08.webshots.com/image/4127/2003057947601864020th.jpg[/img]a)Provethatspacecanbetiledwith1−crosses.b)Provethatspacecanbetiledwith2−crosses.c)Provethatfork≥5 space can not be tiled with k-crosses.

5 A calculating ruler is a ruler for doing algebric calculations. This ruler has three arms, two of themare sationary and one can move freely right and left. Each of arms is gradient. Gradation of eacharm depends on the algebric operation ruler does. For eaxample the ruler below is designed for mul-tiplying two numbers. Gradations are logarithmic. [img]http://aycu05.webshots.com/image/5604/2000468517162383885rs.jpg[/img]Forworkingwithruler, (e.gforcalculatingx.y)wemustmovethemiddlearmthatthearrowatthebeginningofitsgradationlocateabovethexinthelowerarm.Wefindyinthemiddlearm, andwewillreadthenumberontheupperarm.Thenumberwrittenontheruleristheanswer.1)Designarulerforcalculatingxy.Grade first arm (x) and (y) from 1 to 10. 2) Find all rulers that do the multiplication in the interval[1, 10]. 3) Prove that there is not a ruler for calculating x2 + xy + y2, that its first and second armare grade from 0 to 10.

6 Assume that C is a convex subset of Rd. Suppose that C1, C2, . . . , Cn are translations of C thatCi ∩ C 6= ∅ but Ci ∩ Cj = ∅. Prove that

n ≤ 3d − 1













2006

Prove that 3d − 1 is the best bound. P.S. In the exam problem was given for n = 3.

7 We have finite number of distinct shapes in plane. A ”convex Kearting” of these shapes is coveringplane with convex sets, that each set consists exactly one of the shapes, and sets intersect at most inborder. [img]http://aycu30.webshots.com/image/4109/2003791140004582959th.jpg[/img]InwhichcaseConvexkeartingispossible?1)Finitedistinctpoints2)Finitedistinctsegments3)FinitedistinctcirclesWemeanatraingleinQn,3 points that are not collinear in Qn

a) Suppose that ABC is triangle in Qn. Prove that there is a triangle A′B′C ′ in Q5 that ∠B′A′C ′ =∠BAC. b) Find a natural m that for each traingle that can be embedded in Qn it can be embeddedin Qm. c) Find a triangle that can be embedded in Qn and no triangle similar to it can be embeddedin Q3. d) Find a natural m′ that for each traingle that can be embedded in Qn then there is atriangle similar to it, that can be embedded in Qm. You must prove the problem for m = 9 andm′ = 6 to get complete mark. (Better results leads to additional mark.)








2006

8 Geometry

1 Prove that in triangle ABC, radical center of its excircles lies on line GI, which G is Centroidof triangle ABC, and I is the incenter.

2 ABC is a triangle and R,Q,P are midpoints of AB,AC,BC. Line AP intersects RQ in Eand circumcircle of ABC in F . T, S are on RP,PQ such that ES ⊥ PQ, ET ⊥ RP . F ′

is on circumcircle of ABC that FF ′ is diameter. The point of intersection of AF ′ and BCis E′. S′, T ′ are on AB,AC that E′S′ ⊥ AB,E′T ′ ⊥ AC. Prove that TS and T ′S′ areperpendicular.

3 In triangle ABC, if L,M,N are midpoints of AB,AC,BC. And H is orthogonal center oftriangle ABC, then prove that

LH2 + MH2 + NH2 ≤ 14(AB2 + AC2 + BC2)

4 Circle Ω(O,R) and its chord AB is given. Suppose C is midpoint of arc AB. X is an arbitrarypoint on the cirlce. Perpendicular from B to CX intersects circle again in D. Perpendicularfrom C to DX intersects circle again in E. We draw three lines `1, `2, `3 from A,B, E parralellto OX,OD,OC. Prove that these lines are concurrent and find locus of concurrncy point.

5 M is midpoint of side BC of triangle ABC, and I is incenter of triangle ABC, and T ismidpoint of arc BC, that does not contain A. Prove that

cos B + cos C = 1 ⇐⇒ MI = MT













2006

Linear Algebra

1 Suppose that A ∈ Mn(R) with Rank(A) = k. Prove that A is sum of k matrices X1, . . . , Xk

with Rank(Xi) = 1.

2 f : Rn −→ Rm is a non-zero linear map. Prove that there is a base v1, . . . , vnm for Rn thatthe set f(v1), . . . , f(vn) is linearly independent, after ommitting Repetitive elements.

3 Suppose (u, v) is an inner product on Rn and f : Rn −→ Rn is an isometry, that f(0) = 0. 1)Prove that for each u, v we have (u, v) = (f(u), f(v) 2) Prove that f is linear.

4 f : Rn −→ Rn is a bijective map, that Image of every n − 1-dimensional affine space is an− 1-dimensional affine space. 1) Prove that Image of every line is a line. 2) Prove that f isan affine map. (i.e. f = goh that g is a translation and h is a linear map.)











2006

Number Theory

1 n is a natural number. d is the least natural number that for each a that gcd(a, n) = 1 weknow ad ≡ 1 (mod n). Prove that there exist a natural number that ordnb = d

2 n is a natural number that xn+1x+1 is irreducible over Z2[x]. Consider a vector in Zn

2 that it hasodd number of 1’s (as entries) and at least one of its entries are 0. Prove that these vectorand its translations are a basis for Zn

2

3 L is a fullrank lattice in R2 and K is a sub-lattice of L, that A(K)A(L) = m. If m is the least

number that for each x ∈ L, mx is in K. Prove that there exists a basis x1, x2 for L thatx1,mx2 is a basis for K.

4 a, b, c, t are antural numbers and k = ct and n = ak − bk. a) Prove that if k has at least qdifferent prime divisors, then n has at least qt different prime divisors. b)Prove that ϕ(n) iddivisible by 2

t2

5 For each n, suppsoe L(n) is the number of natural numbers 1 ≤ a ≤ n that n|an − 1. Ifp1, p2, . . . , pk are prime divisors of n, define T (n) as (p1−1)(p2−1) . . . (pk−1). a) Prove thatfor each n ∈ N

n|L(n)T (n)

b)Prove that if gcd(n, T (n)) = 1 then ϕ(n) = L(n)T (n)

6 a) P (x), Q(x) are polynomials with rational coefficients and P (x) is not the zero polynomial.Prove that there exist a non-zero polynomial Q(x) ∈ Q[x] that

P (x)|Q(R(x))

b) P,Q are polynomial with integer coefficients and P is monic. Prove that there exist amonic polynomial Q(x) ∈ Z[x] that

P (x)|Q(R(x))













2007

AlgebraAnalysis

1 Let a, b be two complex numbers. Prove that roots of z4 +az2 +b form a rhombus with originas center, if and only if a2

b is a non-positive real number.

2 a, b, c are three different positive real numbers. Prove that:∣∣∣∣a + b

a− b+

b + c

b− c+

c + a

c− a

∣∣∣∣ > 1

3 Find the largest real T such that for each non-negative real numbers a, b, c, d, e such thata + b = c + d + e:√

a2 + b2 + c2 + d2 + e2 ≥ T (√

a +√

b +√

c +√

d +√

e)2

4 a) Let n1, n2, . . . be a sequence of natural number such that ni ≥ 2 and ε1, ε2, . . . be asequence such that εi ∈ 1, 2. Prove that the sequence:

n1

√ε1 + n2

√ε2 + · · ·+ nk

√εk

is convergent and its limit is in (1, 2]. Define n1√

ε1 + n2√

ε2 + . . . to be this limit. b) Provethat for each x ∈ (1, 2] there exist sequences n1, n2, · · · ∈ N and ni ≥ 2 and ε1, ε2, . . . , suchthat ni ≥ 2 and εi ∈ 1, 2, and x = n1

√ε1 + n2

√ε2 + . . .

5 Prove that for two non-zero polynomials f(x, y), g(x, y) with real coefficients the system:f(x, y) = 0g(x, y) = 0

has finitely many solutions in C2 if and only if f(x, y) and g(x, y) are coprime.












2007

Final Exam

1 Consider two polygons P and Q. We want to cut P into some smaller polygons and putthem together in such a way to obtain Q. We can translate the pieces but we can not rotatethem or reflect them. We call P,Q equivalent if and only if we can obtain Q from P (which isobviously an equivalence relation). [img]http://i3.tinypic.com/4lrb43k.png[/img] a) Let P,Qbe two rectangles with the same area(their sides are not necessarily parallel). Prove thatP and Q are equivalent. b) Prove that if two triangles are not translation of each other,they are not equivalent. c) Find a necessary and sufficient condition for polygons P,Q to beequivalent.

2 We call the mapping ∆ : Z\0 −→ N, a degree mapping if and only if for each a, b ∈ Z suchthat b 6= 0 and b 6 |a there exist integers r, s such that a = br + s, and ∆(s) < ∆(b). a) Provethat the following mapping is a degree mapping:

δ(n) = Number of digits in the binary representation of n

b) Prove that there exist a degree mapping ∆0 such that for each degree mapping ∆ and foreach n 6= 0, ∆0(n) ≤ ∆(n). c) Prove that δ = ∆0 [img]http://i16.tinypic.com/4qntmd0.png[/img]

3 We call a set A a good set if it has the following properties: 1. A consists circles in plane. 2.No two element of A intersect. Let A,B be two good sets. We say A,B are equivalent if we canreach from A to B by moving circles in A, making them bigger or smaller in such a way thatduring these operations each circle does not intersect with other circles. Let an be the numberof inequivalent good subsets with n elements. For example a1 = 1, a2 = 2, a3 = 4, a4 = 9.[img]http://i5.tinypic.com/4r0x81v.png[/img] If there exist a, b such that Aan ≤ an ≤ Bbn,we say growth ratio of an is larger than a and is smaller than b. a) Prove that growth ratio ofan is larger than 2 and is smaller than 4. b) Find better bounds for upper and lower growthratio of an.

4 In the following triangular lattice distance of two vertices is length of the shortest path betweenthem. Let A1, A2, . . . , An be constant vertices of the lattice. We want to find a vertex in thelattice whose sum of distances from vertices is minimum. We start from an arbitrary vertex.At each step we check all six neighbors and if sum of distances from vertices of one of theneighbors is less than sum of distances from vertices at the moment we go to that neighbor.If we have more than one choice we choose arbitrarily, as seen in the attached picture.

Obviusly the algorithm finishes a) Prove that when we can not make any move we havereached to the problem’s answer. b) Does this algorithm reach to answer for each connectedgraph?











2007

5 Look at these fractions. At firs step we have 01 and 1

0 , and at each step we write a+bc+d between

ab and c

d , and we do this forever

01

10

01

11

10

01

12

11

21

10

01

13

12

23

11

32

21

31

10

. . .

a) Prove that each of these fractions is irreducible. b) In the plane we have put infinitelymany circles of diameter 1, over each integer on the real line, one circle. The inductivelywe put circles that each circle is tangent to two adjacent circles and real line, and wedo this forever. Prove that points of tangency of these circles are exactly all the num-bers in part a(except 1

0). [img]http://i2.tinypic.com/4m8tmbq.png[/img] c) Prove that inthese two parts all of positive rational numbers appear. If you don’t understand the num-bers, look at [url=http://upload.wikimedia.org/wikipedia/commons/2/21/Arabicnumerals−en.svg]here[/url].Scientisthavesucceededtofindnewnumbersbetweenrealnumberswithstrongmicroscopes.Nowrealnumbersareextendedinanewlargersystemwehaveanorderonit(whichifinducesnormalorderonR), andalso4operationsaddition,multiplication, ...andtheseoperationhaveallpropertiesthesameasR.[img]http : //i14.tinypic.com/4tk6mnr.png[/img]a)Provethatinthislargersystemthereisanumberwhichissmallerthaneachpositiveintegerandislargerthanzero.b)ProvethatnoneofthesenumbersarerootofapolynomialinR[x].

67 A ring is the area between two circles with the same center, and width of a ring is the differencebetween the radii of two circles. [img]http://i18.tinypic.com/6cdmvi8.png[/img] a) Can we putuncountable disjoint rings of width 1(not necessarily same) in the space such that each two of themcan not be separated. [img]http://i19.tinypic.com/4qgx30j.png[/img] b) What’s the answer if 1 isreplaced with 0?

8 In this question you must make all numbers of a clock, each with using 2, exactly 3 times andMathematical symbols. You are not allowed to use English alphabets and words like sin or lim ora, b and no other digits. [img]http://i2.tinypic.com/5x73dza.png[/img]











2007

Geometry

1 Let ABC, l and P be arbitrary triangle, line and point. A′, B′, C ′ are reflections of A,B, Cin point P . A′′ is a point on B′C ′ such that AA′′ ‖ l. B′′, C ′′ are defined similarly. Provethat A′′, B′′, C ′′ are collinear.

2 a) Let ABC be a triangle, and O be its circumcenter. BO and CO intersect with AC,AB atB′, C ′. B′C ′ intersects the circumcircle at two points P,Q. Prove that AP = AQ if and onlyif ABC is isosceles. b) Prove the same statement if O is replaced by I, the incenter.

3 Let I be incenter of triangel ABC, M be midpoint of side BC, and T be the intersection pointof IM with incircle, in such a way that I is between M and T . Prove that ∠BIM−∠CIM =32(∠B − ∠C), if and only if AT ⊥ BC.

4 Let ABC be a triangle, and D be a point where incircle touches side BC. M is midpointof BC, and K is a point on BC such that AK ⊥ BC. Let D′ be a point on BC such thatD′MD′K = DM

DK . Define ωa to be circle with diameter DD′. We define ωB,C similarly. Prove thatevery two of these circles are tangent.

5 Let ABC be a triangle. Squares ABcBaC, CAbAcB and BCaCbA are outside the triangle.Square BcB

′cB

′aBa with center P is outside square ABcBaC. Prove that BP, CaBa and AcBc

are concurrent.












2007

Number Theory

1 Let n be a natural number, such that (n, 2(21386−1)) = 1. Let a1, a2, . . . , aϕ(n) be a reducedresidue system for n. Prove that:

n|a13861 + a1386

2 + · · ·+ a1386ϕ(n)

2 Let m,n be two integers such that ϕ(m) = ϕ(n) = c. Prove that there exist natural numbersb1, b2, . . . , bc such that b1, b2, . . . , bc is a reduced residue system with both m and n.

3 Let n be a natural number, and n = 22007k + 1, such that k is an odd number. Prove that

n 6 |2n−1 + 1

4 Find all integer solutions ofx4 + y2 = z4

5 A hyper-primitive root is a k-tuple (a1, a2, . . . , ak) and (m1,m2, . . . ,mk) with the followingproperty: For each a ∈ N, that (a,m) = 1, has a unique representation in the following form:

a ≡ aα11 aα2

2 . . . aαkk (mod m) 1 ≤ αi ≤ mi

Prove that for each m we have a hyper-primitive root.

6 Something related to this [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=845756845756]problem[/url]:Prove that for a set S ⊂ N, there exists a sequence ai∞i=0 in S such that for each n,

∑ni=0 aix

i

is irreducible in Z[x] if and only if |S| ≥ 2.

By Omid Hatami













2008

Algebra

1 Suppose that f(x) ∈ Z[x] be an irreducible polynomial. It is known that f has a root of normlarger than 3

2 . Prove that if α is a root of f then f(α3 + 1) 6= 0.

2 Find the smallest real K such that for each x, y, z ∈ R+:

x√

y + y√

z + z√

x ≤ K√

(x + y)(y + z)(z + x)

3 Let (b0, b1, b2, b3) be a permutation of the set 54, 72, 36, 108. Prove that x5 + b3x3 + b2x

2 +b1x + b0 is irreducible in Z[x].

4 Let x, y, z ∈ R+ and x + y + z = 3. Prove that:

x3

y3 + 8+

y3

z3 + 8+

z3

x3 + 8≥ 1

9+

227

(xy + xz + yz)

5 Prove that the following polynomial is irreducible in Z[x, y]:

x200y5 + x51y100 + x106 − 4x100y5 + x100 − 2y100 − 2x6 + 4y5 − 2












2008

Combinatorics

1 Prove that the number of pairs (α, S) of a permutation α of 1, 2, . . . , n and a subset S of1, 2, . . . , n such that

∀x ∈ S : α(x) 6∈ S

is equal to n!Fn+1 in which Fn is the Fibonacci sequence such that F1 = F2 = 1

2 Prove that the number permutations α of 1, 2, . . . , n s.t. there does not exist i < j < n s.t.α(i) < α(j + 1) < α(j) is equal to the number of partitions of that set.

3 Prove that for each n:n∑

k=1

(n + k − 12k − 1

)= F2n

4 Let S be a sequence that: S0 = 0S1 = 1

Sn = Sn−1 + Sn−2 + Fn (n > 1)

such that Fn is Fibonacci sequence such that F1 = F2 = 1. Find Sn in terms of Fibonaccinumbers.

5 n people decide to play a game. There are n−1 ropes and each of its two ends are in hand ofone of the players, in such a way that ropes and players form a tree. (Each person can holdmore than rope end.)

At each step a player gives one of the rope ends he is holding to another player. The goal isto make a path of length n− 1 at the end.

But the game regulations change before game starts. Everybody has to give one of his ropeends only two one of his neighbors. Let a and b be minimum steps for reaching to goal inthese two games. Prove that a = b if and only if by removing all players with one rope end(leaves of the tree) the remaining people are on a path. (the remaining graph is a path.)[img]http://i37.tinypic.com/2l9h1tv.png[/img]












2008

Complex numbers

1 Prove that for n > 0 and a 6= 0 the polynomial p(z) = az2n+1 + bz2n + bz + a has a root onunit circle

2 Let g, f : C −→ C be two continuous functions such that for each z 6= 0, g(z) = f(1z ). Prove

that there is a z ∈ C such that f(1z ) = f(−z)

3 For each c ∈ C, let fc(z, 0) = z, and fc(z, n) = fc(z, n − 1)2 + c for n ≥ 1. a) Prove thatif |c| ≤ 1

4 then there is a neighborhood U of origin such that for each z ∈ U the sequencefc(z, n), n ∈ N is bounded. b) Prove that if c > 1

4 is a real number there is a neighborhood Uof origin such that for each z ∈ U the sequence fc(z, n), n ∈ N is unbounded.










2008

Final Exam

1 Police want to arrest on the famous criminals of the country whose name is Kaiser. Kaiseris in one of the streets of a square shaped city with n vertical streets and n horizon-tal streets. In the following cases how many police officers are needed to arrest Kaiser?[img]http://i38.tinypic.com/2i1icecth.png[/img][img]http : //i34.tinypic.com/28rk4s3th.png[/img]a)EachpoliceofficerhasthesamespeedasKaiserandeverypoliceofficerknowsthelocationofKaiseranytime.b)Kaiserhasaninfinitespeed(finitebutwithnobound)andpoliceofficerscanonlyknowwhereheisonlywhenoneofthemseeKaiser.Everybodyinthisproblem(includingpoliceofficersandKaiser)movecontinuouslyandcanstoporchangehispath.Considersixarbitrarypointsinspace.Everytwopointsarejoinedbyasegment.Provethattherearetwotrianglesthatcannotbeseparated.[img]http ://i38.tinypic.com/35n615y.png[/img]

23 a) Prove that there are two polynomials in Z[x] with at least one coefficient larger than 1387 suchthat coefficients of their product is in the set −1, 0, 1. b) Does there exist a multiple of x2−3x+1such that all of its coefficient are in the set −1, 0, 1

4 =A subset S of R2 is called an algebraic set if and only if there is a polynomial p(x, y) ∈ R[x, y]such that

S = (x, y) ∈ R2|p(x, y) = 0

Are the following subsets of plane an algebraic sets? 1. A square [img]http://i36.tinypic.com/28uiaep.png[/img]2. A closed half-circle [img]http://i37.tinypic.com/155m155.png[/img]

5 a) Suppose that RBR′B′ is a convex quadrilateral such that vertices R and R′ have red colorand vertices B and B′ have blue color. We put k arbitrary points of colors blue and red in thequadrilateral such that no four of these k + 4 point (except probably RBR′B′) lie one a circle.Prove that exactly one of the following cases occur? 1. There is a path from R to R′ such thatdistance of every point on this path from one of red points is less than its distance from all bluepoints. 2. There is a path from B to B′ such that distance of every point on this path from one ofblue points is less than its distance from all red points. We call these two paths the blue path andthe red path respectively.

Let n be a natural number. Two people play the following game. At each step one player puts apoint in quadrilateral satisfying the above conditions. First player only puts red point and secondplayer only puts blue points. Game finishes when every player has put n points on the plane. Firstplayer’s goal is to make a red path from R to R′ and the second player’s goal is to make a bluepath from B to B′. b) Prove that if RBR′B′ is rectangle then for each n the second player wins.c) Try to specify the winner for other quadrilaterals.

6 There are five research labs on Mars. Is it always possible to divide Mars to five connected congruentregions such that each region contains exactly on research lab. [img]http://i37.tinypic.com/f2iq8g.png[/img]

7 A graph is called a self-interesting graph if and only if it is isomorphic to a graph whose ev-ery edge is a segment and every two edges intersect. Notice that no edge contains a vertexexcept its two endings. a) Find all n’s for which the cycle of length n is self-intersecting. b)














2008

Prove that in a self-intersecting graph |E(G)| ≤ |V (G)|. c) Find all self-intersecting graphs.[img]http://i35.tinypic.com/x43s5u.png[/img]

8 In an old script found in ruins of Perspolis is written: [code] This script has been finished in a yearwhose 13th power is 258145266804692077858261512663 You should know that if you are skilled inArithmetics you will know the year this script is finished easily.[/code] Find the year the script isfinished. Give a reason for your answer.








2008

Geometry

1 Let ABC be a triangle with BC > AC > AB. Let A′, B′, C ′ be feet of perpendiculars fromA,B, C to BC, AC,AB, such that AA′ = BB′ = CC ′ = x. Prove that: a) If ABC ∼ A′B′C ′

then x = 2r b) Prove that if A′, B′ and C ′ are collinear, then x = R + d or x = R− d.

(In this problem R is the radius of circumcircle, r is radius of incircle and d = OI)

2 Let la, lb, lc be three parallel lines passing through A,B, C respectively. Let l′a be reflectionof la into BC. l′b and l′c are defined similarly. Prove that l′a, l

′b, l

′c are concurrent if and only if

la is parallel to Euler line of triangle ABC.

3 Let ABCD be a quadrilateral, and E be intersection points of AB,CD and AD,BC respec-tively. External bisectors of DAB and DCB intersect at P , external bisectors of ABC andADC intersect at Q and external bisectors of AED and AFB intersect at R. Prove thatP,Q,R are collinear.

4 Let ABC be an isosceles triangle with AB = AC, and D be midpoint of BC, and E befoot of altitude from C. Let H be orthocenter of ABC and N be midpoint of CE. ANintersects with circumcircle of triangle ABC at K. The tangent from C to circumcircle ofABC intersects with AD at F . Suppose that radical axis of circumcircles of CHA and CKFis BC. Find ∠BAC.

5 Let D,E, F be tangency point of incircle of triangle ABC with sides BC, AC,AB. DE andDF intersect the line from A parallel to BC at K and L. Prove that the Euler line of triangleDKL passes through Feuerbach point of triangle ABC.












2008

Number Theory

1 Let k > 1 be an integer. Prove that there exists infinitely many natural numbers such as nsuch that:

n|1n + 2n + · · ·+ kn

2 Prove that there exists infinitely many primes p such that:

13|p3 + 1

3 Let P be a regular polygon. A regular sub-polygon of P is a subset of vertices of P with atleast two vertices such that divides the circumcircle to equal arcs. Prove that there is a subsetof vertices of P such that its intersection with each regular sub-polygon has even number ofvertices.

4 Let u be an odd number. Prove that 33u−13u−1 can be written as sum of two squares.

5 Find all polynomials f ∈ Z[x] such that for each a, b, x ∈ N

a + b + c|f(a) + f(b) + f(c)












2010

Day 1

1 suppose that polynomial p(x) = x2010 ± x2009 ± ...± x± 1 does not have a real root. what isthe maximum number of coefficients to be −1?(14 points)

2 a, b, c are positive real numbers. prove the following inequality:1a2 + 1

b2+ 1

c2+ 1

(a+b+c)2≥ 7

25( 1a + 1

b + 1c + 1

a+b+c)2

(20 points)

3 prove that for each natural number n there exist a polynomial with degree 2n + 1 withcoefficients in Q[x] such that it has exactly 2 complex zeros and it’s irreducible in Q[x].(20points)

4 For each polynomial p(x) = anxn + an−1xn−1 + ... + a1x + a0 we define it’s derivative as this

and we show it by p′(x):

p′(x) = nanxn−1 + (n− 1)an−1xn−2 + ... + 2a2x + a1

a) For each two polynomials p(x) and q(x) prove that:(3 points)

(p(x)q(x))′ = p′(x)q(x) + p(x)q′(x)

b) Suppose that p(x) is a polynomial with degree n and x1, x2, ..., xn are it’s zeros. provethat:(3 points)

p′(x)p(x)

=n∑

i=1

1x− xi

c) p(x) is a monic polynomial with degree n and z1, z2, ..., zn are it’s zeros such that:

|z1| = 1, ∀i ∈ 2, .., n : |zi| ≤ 1

Prove that p′(x) has at least one zero in the disc with length one with the center z1 in complexplane. (disc with length one with the center z1 in complex plane: D = z ∈ C : |z − z1| ≤1)(20 points)

5 x, y, z are positive real numbers such that xy+yz+zx = 1. prove that: 3−√

3+ x2

y + y2

z + z2

x ≥(x + y + z)2 (20 points)

the exam time was 6 hours.












2010

Day 2

1 suppose that a = 3100 and b = 5454. how many zs in [1, 399) exist such that for every c thatgcd(c, 3) = 1, two equations xz ≡ c and xb ≡ c (mod a) have the same number of answers?(100

6points)

2 R is a ring such that xy = yx for every x, y ∈ R and if ab = 0 then a = 0 or b = 0. if for everyIdeal I ⊂ R there exist x1, x2, .., xn in R (n is not constant) such that I = (x1, x2, ..., xn),prove that every element in R that is not 0 and it’s not a unit, is the product of finiteirreducible elements.(100

6 points)

3 If p is a prime number, what is the product of elements like g such that 1 ≤ g ≤ p2 and g isa primitive root modulo p but it’s not a primitive root modulo p2, modulo p2?(100

6 points)

4 sppose that σk : N −→ R is a function such that σk(n) =∑

d|n dk. ρk : N −→ R is a functionsuch that ρk ∗ σk = δ. find a formula for ρk.(100

6 points)

5 prove that if p is a prime number such that p = 12k + 2, 3, 5, 7, 8, 11(k ∈ N ∪ 0), thereexist a field with p2 elements.(100

6 points)

6 g and n are natural numbers such that gcd(g2 − g, n) = 1 and A = gi|i ∈ N and B = x ≡(n)|x ∈ A(by x ≡ (n) we mean a number from the set 0, 1, ..., n − 1 which is congruentwith x modulo n). if for 0 ≤ i ≤ g − 1 ai = |[ni

g , n(i+1)g ) ∩B| prove that g − 1|

∑g−1i=0 iai.( the

symbol | | means the number of elements of the set)(1006 points)

the exam time was 4 hours













2010

Day 3

1 1. In a triangle ABC, O is the circumcenter and I is the incenter. X is the reflection of Ito O. A1 is foot of the perpendicular from X to BC. B1 and C1 are defined similarly. provethat AA1,BB1 and CC1 are concurrent.(12 points)

2 in a quadrilateral ABCD, E and F are on BC and AD respectively such that the area oftriangles AED and BCF is 4

7 of the area of ABCD. R is the intersection point of digonalsof ABCD. AR

RC = 35 and BR

RD = 56 . a) in what ratio does EF cut the digonals?(13 points) b)

find AFFD .(5 points)

3 in a quadrilateral ABCD digonals are perpendicular to each other. let S be the intersectionof digonals. K,L,M and N are reflections of S to AB,BC,CD and DA. BN cuts thecircumcircle of SKN in E and BM cuts the circumcircle of SLM in F . prove that EFLKis concyclic.(20 points)

4 in a triangle ABC, I is the incenter. BI and CI cut the circumcircle of ABC at E and Frespectively. M is the midpoint of EF . C is a circle with diameter EF . IM cuts C at twopoints L and K and the arc BC of circumcircle of ABC (not containing A) at D. prove thatDLIL = DK

IK .(25 points)

5 In a triangle ABC, I is the incenter. D is the reflection of A to I. the incircle is tangent toBC at point E. DE cuts IG at P (G is centroid). M is the midpoint of BC. prove that a)AP ||DM .(15 points) b) AP = 2DM . (10 points)

6 In a triangle ABC, ∠C = 45. AD is the altitude of the triangle. X is on AD such that∠XBC = 90 − ∠B (X is in the triangle). AD and CX cut the circumcircle of ABC in Mand N respectively. if tangent to circumcircle of ABC at M cuts AN at P , prove that P ,Band O are collinear.(25 points)

the exam time was 4 hours and 30 minutes.













2010

Day 4

1 suppose that F ⊆ X(k) and |X| = n. we know that for every three distinct elements ofF like A,B, C, at most one of A ∩ B,B ∩ C and C ∩ A is φ. for k ≤ n

2 prove that: a)

|F| ≤ max(1, 4 − nk ) ×

(n− 1k − 1

).(15 points) b) find all cases of equality in a) for k ≤ n

3 .(5

points)

2 suppose that F ⊆⋃n

j=k+1 X(j) and |X| = n. we know that F is a sperner family and it’s alsoHk. prove that:

∑B∈F

1(n− 1|B| − 1

) ≤ 1 (15 points)

3 suppose that F ⊆ p(X) and |X| = n. we know that for every Ai, Aj ∈ F that Ai ⊇ Aj we

have 3 ≤ |Ai| − |Aj |. prove that: |F| ≤ b2n

3 + 12

(n

bn2 c

)c (20 points)

4 suppose that F ⊆ X(K) and |X| = n. we know that for every three distinct elements of Flike A,B and C we have A ∩B 6⊂ C.

a)(10 points) Prove that :

|F| ≤(

k

bk2c

)+ 1

b)(15 points) if elements of F do not necessarily have k elements, with the above conditionsshow that:

|F| ≤(

n

dn−23 e

)+ 2

5 suppose that F ⊆ p(X) and |X| = n. prove that if |F| >∑k−1

i=0

(n

i

)then there exist Y ⊆ X

with |Y | = k such that p(Y ) = F ∩ Y that F ∩ Y = F ∩ Y : F ∈ F(20 points) you cansee this problem also here: COMBINATORIAL PROBLEMS AND EXERCISES-SECONDEDITION-by LASZLO LOVASZ-AMS CHELSEA PUBLISHING- chapter 13- problem 10(c)!!!

6 Suppose that X is a set with n elements and F ⊆ X(k) and X1, X2, ..., Xs is a partition of X.we know that for every A,B ∈ F and every 1 ≤ j ≤ s, E = B∩(

⋃ji=1 Xi) 6= A∩(

⋃ji=1 Xi) = F

shows that non of E,F have the other one. prove that:

|F| ≤ maxPSi=1 wi=k

s∏i=1

(|Xi|wi

)(15 points)

the exam time was 5 hours and 20 minutes.













2010

Day 5

1 two variable ploynomial

P (x, y) is a two variable polynomial with real coefficients. degree of a monomial means sumof the powers of x and y in it. we denote by Q(x, y) sum of monomials with the most degreein P (x, y). (for example if P (x, y) = 3x4y−2x2y3+5xy2+x−5 then Q(x, y) = 3x4y−2x2y3.)suppose that there are real numbers x1,y1,x2 and y2 such that Q(x1, y1) > 0 , Q(x2, y2) < 0prove that the set (x, y)|P (x, y) = 0 is not bounded. (we call a set S of plane bounded ifthere exist positive number M such that the distance of elements of S from the origin is lessthan M .)

time allowed for this question was 1 hour.

2 rolling cube

a,b and c are natural numbers. we have a (2a + 1) × (2b + 1) × (2c + 1) cube. this cube ison an infinite plane with unit squares. you call roll the cube to every side you want. faces ofthe cube are divided to unit squares and the square in the middle of each face is coloured (itmeans that if this square goes on a square of the plane, then that square will be coloured.)prove that if any two of lengths of sides of the cube are relatively prime, then we can colourevery square in plane.


3 points in plane

set A containing n points in plane is given. a copy of A is a set of points that is made byusing transformation, rotation, homogeneity or their combination on elements of A. we wantto put n copies of A in plane, such that every two copies have exactly one point in commonand every three of them have no common elements. a) prove that if no 4 points of A makea parallelogram, you can do this only using transformation. (A doesn’t have a parallelogramwith angle 0 and a parallelogram that it’s two non-adjacent vertices are one!) b) prove thatyou can always do this by using a combination of all these things.

time allowed for this question was 1 hour and 30 minutes

4 carpeting

suppose that S is a figure in the plane such that it’s border doesn’t contain any lattice points.suppose that x, y are two lattice points with the distance 1 (we call a point lattice point ifit’s coordinates are integers). suppose that we can cover the plane with copies of S such thatx, y always go on lattice points ( you can rotate or reverse copies of S). prove that the areaof S is equal to lattice points inside it.












2010

5 interesting sequence

n is a natural number and x1, x2, ... is a sequence of numbers 1 and −1 with these properties:

it is periodic and its least period number is 2n − 1. (it means that for every natural numberj we have xj+2n−1 = xj and 2n − 1 is the least number with this property.)

There exist distinct integers 0 ≤ t1 < t2 < ... < tk < n such that for every natural number jwe have

xj+n = xj+t1 × xj+t2 × ...× xj+tk

Prove that for every natural number s that s < 2n − 1 we have

2n−1∑i=1

xixi+s = −1

Time allowed for this question was 1 hours and 15 minutes.

6 polyhedral

we call a 12-gon in plane good whenever: first, it should be regular, second, it’s inner planemust be filled!!, third, it’s center must be the origin of the coordinates, forth, it’s verticesmust have points (0, 1),(1, 0),(−1, 0) and (0,−1). find the faces of the massivest polyhedralthat it’s image on every three plane xy,yz and zx is a good 12-gon. (it’s obvios that centersof these three 12-gons are the origin of coordinates for three dimensions.)

time allowed for this question is 1 hour.

7 interesting function

S is a set with n elements and P (S) is the set of all subsets of S and f : P (S) → N is afunction with these properties: for every subset A of S we have f(A) = f(S − A). for everytwo subsets of S like A and B we have max(f(A), f(B)) ≥ f(A ∪ B) prove that number ofnatural numbers like x such that there exists A ⊆ S and f(A) = x is less than n.

time allowed for this question was 1 hours and 30 minutes.

8 numbers n2 + 1

prove that there infinitly many natural numbers in the form n2 +1 such that they don’t haveany divider in the form of k2 + 1 except 1 and itself.

time allowed for this question was 45 minutes.











2010

Day 6

1 prove that the group of oriention-preserving symmetries of a cube is isomorph to S4(group ofpermutations of 1, 2, 3, 4).(20 points)

2 prove the third sylow theorem: suppose that G is a group and |G| = pem which p is a primenumber and (p, m) = 1. suppose that a is the number of p-sylow subgroups of G (H < G that|H| = pe). prove that a|m and p|a− 1.(Hint: you can use this: every two p-sylow subgroupsare conjugate.)(20 points)

3 suppose that G < Sn is a subgroup of permutations of 1, ..., n with this property that forevery e 6= g ∈ G there exist exactly one k ∈ 1, ..., n such that g.k = k. prove that thereexist one k ∈ 1, ..., n such that for every g ∈ G we have g.k = k.(20 points)

4 a) prove that every discrete subgroup of (R2,+) is in one of these forms: i-0. ii-mv|m ∈ Zfor a vector v in R2. iii-mv + nw|m,n ∈ Z for tho linearly independent vectors v and win R2.(lattice L) b) prove that every finite group of symmetries that fixes the origin and thelattice L is in one of these forms: Ci or Di that i = 1, 2, 3, 4, 6 (Ci is the cyclic group of orderi and Di is the dyhedral group of order i).(20 points)

5 suppose that p is a prime number. find that smallest n such that there exists a non-abeliangroup G with |G| = pn.

SL is an acronym for Special Lesson. this year our special lesson was Groups and Symmetries.

the exam time was 5 hours.











iranTeam Selection Test

2002

1 ABCD is a convex quadrilateral. We draw its diagnals to divide the quadrilateral to four tri-abgles. P is the intersection of diagnals. I1, I2, I3, I4 are excenters of PAD,PAB,PBC, PCD(excenterscorresponding vertex P ). Prove that I1, I2, I3, I4 lie on a circle iff ABCD is a tangentialquadrilateral.

2 n people (with names 1, 2, . . . , n) are around a table. Some of them are friends. At each step 2friend can change their place. Find a necessary and sufficient condition for friendship relationbetween them that with these steps we can always reach to all of posiible permutations.

3 A ”2-line” is the area between two parallel lines. Length of ”2-line” is distance of two parallellines. We have covered unit circle with some ”2-lines”. Prove sum of lengths of ”2-lines” isat least 2.

4 O is a point in triangle ABC. We draw perpendicular from O to BC, AC,AB which intersectBC, AC,AB at A1, B1, C1. Prove that O is circumcenter of triangle ABC iff perimeter ofABC is not less than perimeter of triangles AB1C1, BC1A1, CB1A1.

5 A school has n students and k classes. Every two students in the same class are friends. Foreach two different classes, there are two people from these classes that are not friends. Provethat we can divide students into n− k + 1 parts taht students in each part are not friends.

6 Assume x1, x2, . . . , xn ∈ R+,∑n

i=1 x2i = n,

∑ni=1 xi ≥ s > 0 and 0 ≤ λ ≤ 1. Prove that at

least⌈

s2(1−λ)2

n

⌉of these numbers are larger than λs

n .

7 S1, S2, S3 are three spheres in R3 that their centers are not collinear. k ≤ 8 is the numberof planes that touch three spheres. Ai, Bi, Ci is the point that i-th plane touch the spheresS1, S2, S3. Let Oi be circumcenter of AiBiCi. Prove that Oi are collinear.

8 We call A1, A2, A3 mangool iff there is a permutation π that Aπ(2) 6⊂ Aπ(1), Aπ(3) 6⊂ Aπ(1) ∪Aπ(2). A good family is a family of finite subsets of N like X, A1, A2, . . . , An. To each goofamily we correspond a graph with vertices A1, A2, . . . , An. Connect Ai, Aj iff X, Ai, Aj aremangool sets. Find all graphs that we can find a good family corresponding to it.

9 π(n) is the number of primes that are not bigger than n. For n = 2, 3, 4, 6, 8, 33, . . . we haveπ(n)|n. Does exist infinitely many integers n that π(n)|n?

10 Suppose from (m + 2)× (n + 2) rectangle we cut 4, 1× 1 corners. Now on first and last rowfirst and last columns we write 2(m + n) real numbers. Prove we can fill the interior m × nrectangle with real numbers that every number is average of it’s 4 neighbors.

11 A 10 × 10 × 10 cube has 1000 unit cubes. 500 of them are coloured black and 500 of themare coloured white. Show that there are at least 100 unit squares, being the common face ofa white and a black unit cube.


















2002

12 We call a permutation (a1, a2, ..., an) of (1, 2, ..., n) quadratic if there exists at least a perfectsquare among the numbers a1, a1 + a2, ..., a1 + a2 + ...+ an. Find all natural numbers n suchthat all permutations in Sn are quadratic.

Remark. Sn denotes the n-th symmetric group, the group of permutations on n elements.

13 Let ABC be a triangle. The incircle of triangle ABC touches the side BC at A′, and theline AA′ meets the incircle again at a point P . Let the lines CP and BP meet the incircle oftriangle ABC again at N and M , respectively. Prove that the lines AA′, BN and CM areconcurrent.









2004

1 Suppose that p is a prime number. Prove that for each k, there exists an n such that:(n

p

)=

(n + k

p

)

2 Suppose that p is a prime number. Prove that the equation x2 − py2 = − 1 has a solution ifand only if p ≡ 1 (mod 4).

3 Suppose that ABCD is a convex quadrilateral. Let F = AB ∩ CD, E = AD ∩ BC andT = AC ∩BD. Suppose that A,B, T, E lie on a circle which intersects with EF at P . Provethat if M is midpoint of AB, then ∠APM = ∠BPT .

4 Let M,M ′ be two conjugates point in triangle ABC (in the sense that ∠MAB = ∠M ′AC, . . . ).Let P,Q,R, P ′, Q′, R′ be foots of perpendiculars from M and M ′ to BC, CA, AB. LetE = QR ∩Q′R′, F = RP ∩R′P ′ and G = PQ ∩ P ′Q′. Prove that the lines AG, BF, CE areparallel.

5 This problem is generalization of [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=5918]thisone[/url]. Suppose G is a graph and S ⊂ V (G). Suppose we have arbitrarily assign real num-bers to each element of S. Prove that we can assign numbers to each vertex in G\S that foreach v ∈ G\S number assigned to v is average of its neighbors.

6 p is a polynomial with integer coefficients and for every natural n we have p(n) > n xk is a sequencethat: x1 = 1, xi+1 = p(xi)for every N one of xi is divisible by N Prove p(x) = x + 1

N opain,nogain













2005

Day 1

1 Suppose that a1, a2, ..., an are positive real numbers such that a1 ≤ a2 ≤ · · · ≤ an. Let

a1 + a2 + · · ·+ an

n= m;

a21 + a2

2 + · · ·+ a2n

n= 1.

Suppose that, for some i, we know ai ≤ m. Prove that:

n− i ≥ n (m− ai)2

2 Assume ABC is an isosceles triangle that AB = AC Suppose P is a point on extension ofside BC. X and Y are points on AB and AC that:

PX||AC , PY ||AB

Also T is midpoint of arc BC. Prove that PT ⊥ XY

3 Suppose there are 18 lighthouses on the Persian Gulf. Each of the lighthouses lightens anangle with size 20 degrees. Prove that we can choose the directions of the lighthouses suchthat whole of the blue Persian (always Persian) Gulf is lightened.










2005

Day 2

1 Find all f : N 7−→ N that there exist k ∈ N and a prime p that: ∀n ≥ k f(n+ p) = f(n) andalso if m | n then f(m + 1) | f(n) + 1

2 Suppose there are n distinct points on plane. There is circle with radius r and center O onthe plane. At least one of the points are in the circle. We do the following instructions. Ateach step we move O to the baricenter of the point in the circle. Prove that location of O isconstant after some steps.

3 Suppose S = 1, 2, . . . , n and n ≥ 3. There is f : Sk 7−→ S that if a, b ∈ Sk and a and bdiffer in all of elements then f(a) 6= f(b). Prove that f is a function of one of its elements.










2006

Day 1

1 Suppose that p is a prime number. Find all natural numbers n such that p|ϕ(n) and for alla such that (a, n) = 1 we have

n|aϕ(n)

p − 1

2 Suppose n coins are available that their mass is unknown. We have a pair of balances andevery time we can choose an even number of coins and put half of them on one side of thebalance and put another half on the other side, therefore a comparison will be done. Our aimis determining that the mass of all coins is equal or not. Show that at least n−1 comparisonsare required.

3 Suppose ABC is a triangle with M the midpoint of BC. Suppose that AM intersects theincircle at K, L. We draw parallel line from K and L to BC and name their second intersectionpoint with incircle X and Y . Suppose that AX and AY intersect BC at P and Q. Provethat BP = CQ.










2006

Day 2

4 Let x1, x2, . . . , xn be real numbers. Prove that

n∑i,j=1

|xi + xj | ≥ n

n∑i=1

|xi|

5 Let ABC be a triangle such that it’s circumcircle radius is equal to the radius of outerinscribed circle with respect to A. Suppose that the outer inscribed circle with respect to Atouches BC, AC,AB at M,N,L. Prove that O (Center of circumcircle) is the orthocenter ofMNL.

6 Let G be a tournoment such that it’s edges are colored either red or blue. Prove that thereexists a vertex of G like v with the property that, for every other vertex u there is a mono-colordirected path from v to u.










2006

Day 3

1 We have n points in the plane, no three on a line. We call k of them good if they form aconvex polygon and there is no other point in the convex polygon. Suppose that for a fixedk the number of k good points is ck. Show that the following sum is independent of thestructure of points and only depends on n :

n∑i=3

(−1)ici

2 Let n be a fixed natural number. a) Find all solutions to the following equation :

n∑k=1

[x

2k] = x− 1

b) Find the number of solutions to the following equation (m is a fixed natural) :

n∑k=1

[x

2k] = x−m

3 Let l, m be two parallel lines in the plane. Let P be a fixed point between them. Let E,F bevariable points on l,m such that the angle EPF is fixed to a number like α where 0 < α < π

2 .(By angle EPF we mean the directed angle) Show that there is another point (not P ) suchthat it sees the segment EF with a fixed angle too.










2006

Day 4

4 Let n be a fixed natural number. Find all n tuples of natural pairwise distinct and coprimenumbers like a1, a2, . . . , an such that for 1 ≤ i ≤ n we have

a1 + a2 + . . . + an|ai1 + ai

2 + . . . + ain

5 Let ABC be an acute angle triangle. Suppose that D,E, F are the feet of perpendicluar linesfrom A,B, C to BC, CA, AB. Let P,Q,R be the feet of perpendicular lines from A,B, C toEF,FD,DE. Prove that

2(PQ + QR + RP ) ≥ DE + EF + FD

6 Suppose we have a simple polygon (that is it does not intersect itself, but not necessarilyconvex). Show that this polygon has a diameter which is completely inside the polygon andthe two arcs it creates on the polygon perimeter (the two arcs have 2 vertices in common)both have at least one third of the vertices of the polygon.










2007

Day 1

1 In triangle ABC, M is midpoint of AC, and D is a point on BC such that DB = DM . Weknow that 2BC2 −AC2 = AB.AC. Prove that

BD.DC =AC2.AB

2(AB + AC)

2 Let A be the largest subset of 1, . . . , n such that for each x ∈ A, x divides at most oneother element in A. Prove that

2n

3≤ |A| ≤

⌈3n

4

⌉edited by pbornsztein

3 Find all solutions of the following functional equation:

f(x2 + y + f(y)) = 2y + f(x)2










2007

Day 2

1 In an isosceles right-angled triangle shaped billiards table , a ball starts moving from one ofthe vertices adjacent to hypotenuse. When it reaches to one side then it will reflect its path.Prove that if we reach to a vertex then it is not the vertex at initial position

By Sam Nariman

2 Find all monic polynomials f(x) in Z[x] such that f(Z) is closed under multiplication.

By Mohsen Jamali

3 Let ω be incircle of ABC. P and Q are on AB and AC, such that PQ is parallel to BCand is tangent to ω. AB,AC touch ω at F,E. Prove that if M is midpoint of PQ, and T isintersection point of EF and BC, then TM is tangent to ω.

By Ali Khezeli










2007

Day 3

1 1) Does there exist a a sequence a0, a1, a2, . . . in N, such that for each i 6= j, (ai, aj) = 1, andfor each n, the polynomial

∑ni=0 aix

i is irreducible in Z[x]?

By Omid Hatami

2 Suppose n lines in plane are such that no two are parallel and no three are concurrent. Foreach two lines their angle is a real number in [0, π

2 ]. Find the largest value of the sum of the(n2

)angles between line.

By Aliakbar Daemi

3 O is a point inside triangle ABC such that OA = OB + OC. Suppose B′, C ′ be midpoints ofarcs AOC and AOB. Prove that circumcircles COC ′ and BOB′ are tangent to each other.










2007

Day 4

1 Find all polynomials of degree 3, such that for each x, y ≥ 0:

p(x + y) ≥ p(x) + p(y)

2 Triangle ABC is isosceles (AB = AC). From A, we draw a line ` parallel to BC. P,Q areon perpendicular bisectors of AB,AC such that PQ ⊥ BC. M,N are points on ` such thatangles ∠APM and ∠AQN are π

2 . Prove that

1AM

+1

AN≤ 2

AB

3 Let P be a point in a square whose side are mirror. A ray of light comes from P and with slopeα. We know that this ray of light never arrives to a vertex. We make an infinite sequence of0, 1. After each contact of light ray with a horizontal side, we put 0, and after each contactwith a vertical side, we put 1. For each n ≥ 1, let Bn be set of all blocks of length n, in thissequence. a) Prove that Bn does not depend on location of P . b) Prove that if α

π is irrational,then |Bn| = n + 1.










2008

1 Find all functions f : R −→ R such that for each x, y ∈ R:

f(xf(y)) + y + f(x) = f(x + f(y)) + yf(x)

2 Suppose that I is incenter of triangle ABC and l′ is a line tangent to the incircle. Let l beanother line such that intersects AB,AC,BC respectively at C ′, B′, A′. We draw a tangentfrom A′ to the incircle other than BC, and this line intersects with l′ at A1. B1, C1 aresimilarly defined. Prove that AA1, BB1, CC1 are concurrent.

3 Suppose that T is a tree with k edges. Prove that the k-dimensional cube can be partitionedto graphs isomorphic to T .

4 Let P1, P2, P3, P4 be points on the unit sphere. Prove that∑

i6=j1

|Pi−Pj | takes its minimumvalue if and only if these four points are vertices of a regular pyramid.

5 Let a, b, c > 0 and ab + ac + bc = 1. Prove that:√a3 + a +

√b3 + b +

√c3 + c ≥ 2

√a + b + c

6 Prove that in a tournament with 799 teams, there exist 14 teams, that can be partitionedinto groups in a way that all of the teams in the first group have won all of the teams in thesecond group.

7 Let S be a set with n elements, and F be a family of subsets of S with 2n−1 elements, suchthat for each A,B, C ∈ F , A ∩ B ∩ C is not empty. Prove that the intersection of all of theelements of F is not empty.

8 Find all polynomials p of one variable with integer coefficients such that if a and b are naturalnumbers such that a + b is a perfect square, then p (a) + p (b) is also a perfect square.

9 Ia is the excenter of the triangle ABC with respect to A, and AIa intersects the circumcircleof ABC at T . Let X be a point on TIa such that XI2

a = XA.XT . Draw a perpendicular linefrom X to BC so that it intersects BC in A′. Define B′ and C ′ in the same way. Prove thatAA′, BB′ and CC ′ are concurrent.

10 In the triangle ABC, ∠B is greater than ∠C. T is the midpoint of the arc BAC from thecircumcircle of ABC and I is the incenter of ABC. E is a point such that ∠AEI = 90 andAE ‖ BC. TE intersects the circumcircle of ABC for the second time in P . If ∠B = ∠IPB,find the angle ∠A.

11 k is a given natural number. Find all functions f : N → N such that for each m,n ∈ N thefollowing holds:

f(m) + f(n) | (m + n)k


















2008

12 In the acute-angled triangle ABC, D is the intersection of the altitude passing through Awith BC and Ia is the excenter of the triangle with respect to A. K is a point on the extensionof AB from B, for which ∠AKIa = 90 + 3

4∠C. IaK intersects the extension of AD at L.Prove that DIa bisects the angle ∠AIaB iff AL = 2R. (R is the circumradius of ABC)








2009

Day 1

1 Let ABC be a triangle and A′ , B′ and C ′ lie on BC , CA and AB respectively such thatthe incenter of A′B′C ′ and ABC are coincide and the inradius of A′B′C ′ is half of inradiusof ABC . Prove that ABC is equilateral .

2 Let a be a fix natural number . Prove that the set of prime divisors of 22n+a for n = 1, 2, · · ·

is infinite

3 Suppose that a,b,c be three positive real numbers such that a + b + c = 3 . Prove that :1

2+a2+b2+ 1

2+b2+c2+ 1

2+c2+a2 ≤ 34










2009

Day 2

1 Find All Polynomials f with integer coefficient such that , for every prime p and every naturalnumbers u and v with the condition : p|uv − 1 we always have p|f(u)f(v)− 1

2 ABC is a triangle and AA′ , BB′ and CC ′ are three altitudes of this triangle . Let P be thefeet of perpendicular from C ′ to A′B′ , and Q is a point on A′B′ such that QA = QB . Provethat : ∠PBQ = ∠PAQ = ∠PC ′C

3 We have a closed path on a vertices of a nn square which pass from each vertice exactly once. prove that we have two adjacent vertices such that if we cut the path from these pointsthen length of each pieces is not less than quarter of total path .










2009

Day 3

1 Suppose three direction on the plane . We draw 11 lines in each direction . Find maximumnumber of the points on the plane which are on three lines .

2 Find All Polynomials P (x, y) such that for all reals x, y we have : P (x2, y2) = P ( (x+y)2

2 , (x−y)2

2 )

3 In triangle ABC , D , E and F are the points of tangency of incircle with the center of I toBC , CA and AB respectively . Let M be the feet of perpendicular from D to EF and Pis on DM such that : DP = MP . If H be the orthocenter of BIC, prove that PH bisectsEF .










2009

Day 4

1 Let ABC be a triangle and AB 6= AC . D is a point on BC such that BA = BD and B isbetween C and D . Let Ic be center of the circle which touches AB and the extensions of ACand BC . CIc intersect the circumcircle of ABC again at T . If ∠TDIc = ∠B+∠C

4 then find∠A

2 n is a positive integer . prove that :

352

n−1

2n+2 ≡ (−5)32

n−1

2n+2 (mod 2n+4)

3 T is a subset of 1, 2, ..., n which has this property : for all distinct i, j ∈ T , 2j is not divisibleby i . Prove that : |T | ≤ 4

9n + log2n + 2









iranPre-Preparation Course Examination

2004

1 A network is a simple directed graph such that each edge e has two intger lower and uppercapacities 0 ≤ cl(e) ≤ cu(e). A circular flow on this graph is a function such that: 1) For eachedge e, cl(e) ≤ f(e) ≤ cu(e). 2) For each vertex v:∑

e∈v+

f(e) =∑e∈v−

f(e)

a) Prove that this graph has a circular flow, if and only if for each partition X, Y of verticesof the network we have: ∑

e = xyx ∈ X, y ∈ Y

cl(e) ≤∑

e = yxy ∈ Y, x ∈ X

cl(e)

b) Suppose that f is a circular flow in this network. Prove that there exists a circular flow gin this network such that g(e) = bf(e)c or g(e) = df(e)e for each edge e.

2 Let H(n) be the number of simply connected subsets with n hexagons in an infinite hexagonalnetwork. Also let P (n) be the number of paths starting from a fixed vertex (that do notconnect itself) with lentgh n in this hexagonal network. a) Prove that the limits

α := limn→∞

H(n)1n , β := lim

n→∞P (n)

1n

exist.

b) Prove the following inequalities:√

2 ≤ β ≤ 2 α ≤ 12.5 α ≥ 3.5 α ≤ β4

3 For a subset S of vertices of graph G, let Λ(S) be the subset of all edges of G such that at leastone of their ends is in S. Suppose that G is a graph with m edges. Let d∗ : V (G) −→ N∪0be a function such that a)

∑u d∗(u) = m. b) For each subset S of V (G):∑

u∈S

d∗(u) ≤ |Λ(S)|

Prove that we can give directions to edges of G such that for each edge e, d+(e) = d∗(e).

4 Let G be a simple graph. Suppose that size of largest independent set in G is α. Prove that:a) Vertices of G can be partitioned to at most α paths. b) Suppose that a vertex and an edgeare also cycles. Prove that vertices of G can be partitioned to at most α cycles.

5 Let A = A1, . . . , Am be a family distinct subsets of 1, 2, . . . , n with at most n2 elements.

Assume that Ai 6⊂ Aj and Ai ∩Aj 6= ∅ for each i, j. Prove that:m∑

i=1

1(n−1|Ai|−1

) ≤ 1












2004

6 Let l, d, k be natural numbers. We want to prove that for large numbers n, for each k-coloringof the n-dimensional cube with side length l, there is a d-dimensional subspace that all of itsvertices have the same color. Let H(l, d, k) be the least number such that for n ≥ H(l, d, k)the previus statement holds. a) Prove that:

H(l, d + 1, k) ≤ H(l, 1, k) + H(l, d, kl)H(l,1,k)

b) Prove thatH(l + 1, 1, k + 1) ≤ H(l, 1 + H(l + 1, 1, k), k + 1)

c) Prove the statement of problem. d) Prove Van der Waerden’s Theorem.

7 Let G = (V,E) ve a simple graph.

a) Let A,B be a subsets of E, and spanning subgraphs of G with edges A,B, A∪B and A∩Bhave a, b, c and d connected components respectively. Prove that a + b ≤ c + d

We say that subsets A1, A2, . . . , Am of E have (R) property if and only if for each I ⊂1, 2, . . . ,m the spanning subgraph of G with edges ∪i∈IAi has at most n − |I| connectedcomponents. b) Prove that when A1, . . . , Am, B have (R) property, and |B| ≥ 2, there existsan x ∈ B such that A1, A2, . . . , Am, B\x also have property (R).

Suppose that edges of G are colored arbitrarily. A spanning subtree in G is called colorful ifand only if it does not have any two edges with the same color. c) Prove that G has a colorfulsubtree if and only if for each partition of V to k non-empty subsets such as V1, . . . , Vk, thereare at least k − 1 edges with distinct colors that each of these edges has its two ends in twodifferent Vis. d) Assume that edges of Kn has been colored such that each color is repeated[

n2

]times. Prove that there exists a colorful subtree. e) Prove that in part d) if n ≥ 5 there

is a colorful subtree that is non-isomorphic to K1,n−1. f) Prove that in part e) there are atleast two non-intersecting colorful subtrees.









2006

Combinatorics

1 a) Find the value of∑∞

n=1φ(n)2n−1 ;

b) Show that∑

k

(mk

)(n+km

)=

∑k

(mk

)(nk

)2k for m,n ≥ 0;

c) Using the identity (1− x)−12 (1− x)−

12 = (1− x)−1 derive a combinatorial identity!

d) Express the value of∑

(2a1−1) . . . (2ak−1) where the sum is over all 2n−1 ways of choosing(a1, a2, . . . , ak) such that a1 + a2 + . . . + ak = n, as a function of some Fibonacci term.

2 If f(x) is the generating function of the sequence a1, a2, . . . and if f(x) = r(x)s(x) holds such that

r(x) and s(x) are polynomials show that an has a homogenous recurrence.

3 The bell number bn is the number of ways to partition the set 1, 2, . . . , n. For exampleb3 = 5. Find a recurrence for bn and show that bn = e−1

∑k≥0

kn

k! . Using a combinatorialproof show that the number of ways to partition 1, 2, . . . , n, such that now two consecutivenumbers are in the same block, is bn−1.

4 Show that for every prime p and integer n, there is an irreducible polynomial of degree n inZp[x] and use that to show there is a field of size pn.

5 Express the sum Sm(n) = 1m + 2m + . . . + (n− 1)m with Bernolli numbers.

6 Show that the product of every k consecutive members of the Fibonacci sequence is divisibleby f1f2 . . . fk (where f0 = 0 and f1 = 1).

7 Suppose that for every n the number m(n) is chosen such that m(n) ln(m(n)) = n− 12 . Show

that bn is asymptotic to the following expression where bn is the n−th Bell number, that isthe number of ways to partition 1, 2, . . . , n:

m(n)nem(n)−n− 12

√lnn

.

Two functions f(n) and g(n) are asymptotic to each other if limn→∞f(n)g(n) = 1.

8 Suppose that p(n) is the number of ways to express n as a sum of some naturall numbers (thetwo representations 4 = 1 + 1 + 2 and 4 = 1 + 2 + 1 are considered the same). Prove that foran infinite number of n’s p(n) is even and for an infinite number of n’s p(n) is odd.















2006

Dynamical Systems

1 Suppose that X is a compact metric space and T : X → X is a continous function. Provethat T has a returning point. It means there is a strictly increasing sequence ni such thatlimk→∞ Tnk(x0) = x0 for some x0.

2 Show that there exists a continuos function f : [0, 1] → [0, 1] such that it has no periodic orbitof order 3 but it has a periodic orbit of order 5.

3 Show that if f : [0, 1] → [0, 1] is a continous function and it has topological transitivity thenperiodic points of f are dense in [0, 1]. Topological transitivity means there for every opensets U and V there is n > 0 such that fn(U) ∩ V 6= ∅.

4 Show that ρ(f) changes continously over f . It means for every bijection f : S1 → S1 andε > 0 there is δ > 0 such that if g : S1 → S1 is a bijection such that ‖ f − g ‖ < δ then|ρ(f)− ρ(g)| < ε.

Note that ρ(f) is the rotatation number of f and ‖ f − g ‖ = sup|f(x)− g(x)||x ∈ S1.

5 Powers of 2 in base 10 start with 3 or 4 more frequently? What is their state in base 3? Firstwrite down an exact form of the question.

6 Suppose that Pc(z) = z2 + c. You are familiar with the Mandelbrot set: M = c ∈C| limn→∞ Pn

c (0) 6= ∞. We know that if c ∈ M then the points of the dynamical system(C, Pc) that don’t converge to ∞ are connected and otherwise they are completely discon-nected. By seeing the properties of periodic points of Pc prove the following ones:

a) Prove the existance of the heart like shape in the Mandelbrot set. b) Prove the existanceof the large circle next to the heart like shape in the Mandelbrot set.

[img]http://astronomy.swin.edu.au/ pbourke/fractals/mandelbrot/mandel1.gif[/img]













2006

Geometry

1 Show that for a triangle we have

maxama, bmb, cmc ≤ sR

where ma denotes the length of median of side BC and s is half of the perimeter of thetriangle.

2 Using projective transformations prove the Pascal theorem (also find where the Pascal lineintersects the circle).

3 There is a right angle whose vertex moves on a fixed circle and one of it’s sides passes a fixedpoint. What is the curve that the other side of the angle is always tangent to it.

4 Find a 3rd degree polynomial whose roots are ra, rb and rc where ra is the radius of the outerinscribed circle of ABC with respect to A.

5 Suppose ∆ is a fixed line and F and F ′ are two points with equal distance from ∆ that are ontwo sides of ∆. The circle C is with center P and radius mPF where m is a positive numbernot equal to 1. The circle C ′ is the circle that PFF ′ is inscribed in it.

a) What is the condition on P such that C and C ′ intersect?

b) If we denote the intersections of C and C ′ to be M and M ′ then what is the locus of Mand M ′;

c) Show that C is always tangent to this locus.












2006

Superior Algebra

1 Find out wich of the following polynomials are irreducible.

a) t4 + 1 over R;

b) t4 + 1 over Q;

c) t3 − 7t2 + 3t + 3 over Q;

d) t4 + 7 over Z17;

e) t3 − 5 over Z11;

f) t6 + 7 over Q(i).

2 a) Show that you can divide an angle θ to three equal parts using compass and ruler if andonly if the polynomial 4t3 − 3t− cos(θ) is reducible over Q(cos(θ)).

b) Is it always possible to divide an angle into five equal parts?

3 a) If K is a finite extension of the field F and K = F (α, β) show that [K : F ] ≤ [F (α) :F ][F (β) : F ]

b) If gcd([F (α) : F ], [F (β) : F ]) = 1 then does the above inequality always become equality?

c) By giving an example show that if gcd([F (α) : F ], [F (β) : F ]) 6= 1 then equality mighthappen.

4 If d ∈ Q, is there always an ω ∈ C such that ωn = 1 for some n ∈ N and Q(√

d) ⊆ Q(ω)?











2007

Combinatorics

1 a) There is an infinite sequence of 0, 1, like . . . , a−1, a0, a1, . . . (i.e. an element of 0, 1Z).At each step we make a new sequence. There is a function f such that for each i, new ai =f(ai−100, ai−99, . . . , ai+100). This operation is mapping F : 0, 1Z −→ 0, 1Z. Prove that ifF is 1-1, then it is surjective. b) Is the statement correct if we have an fi for each i?

2 There is a WORD game with the following rules. There are finite number of relations Ui −→Vi(Ui, Vi are words). There is are two words A,B. We start from A, and we want to reachto B. At each step we can change one subword Ui to Vi. Prove that there does not exist analgorithm that picks up A,B and Ui’s,Vi’s and decides whether we can reach from A to B ornot.









2007

Geometry

1 D is an arbitrary point inside triangle ABC, and E is inside triangle BDC. Prove that

SDBC

(PDBC)2≥ SEBC

(PEBC)2

2 Let C1, C2 and C3 be three circles that does not intersect and non of them is inside an-other. Suppose (L1, L2), (L3, L4) and (L5, L6) be internal common tangents of (C1, C2),(C1, C3), (C2, C3). Let L1, L2, L3, L4, L5, L6 be sides of polygon AC ′BA′CB′. Prove thatAA′, BB′, CC ′ are concurrent.

3 ABC is an arbitrary triangle. A′, B′, C ′ are midpoints of arcs BC, AC,AB. Sides of triangleABC, intersect sides of triangle A′B′C ′ at points P,Q,R, S, T, F . Prove that

SPQRSTF

SABC= 1− ab + ac + bc

(a + b + c)2

4 Let (C) and (L) be a circle and a line. P1, . . . , P2n+1 are odd number of points on (L). A1 isan arbitrary point on (C). Ak+1 is the intersection point of AkPk and (C) (1 ≤ k ≤ 2n + 1).Prove that A1A2n+2 passes through a constant point while A1 varies on (C).











2007

Number Theory Algebra

1 Let a ≥ 2 be a natural number. Prove that∑∞

n=01

an2 is irrational.

2 Let A1, . . . , Ak be matrices which make a group under matrix multiplication. SupposeM = A1 + · · ·+ Ak. Prove that each eigenvalue of M is equal to 0 or k.

3 This question is both combinatorics and Number Theory : a ) Prove that we can color edgesof Kp with p colors which is proper, (p is an odd prime) and Kp can be partitioned to p−1

2rainbow Hamiltonian cycles. (A Hamiltonian cycle is a cycle that passes from all of verteces,and a rainbow is a subgraph that all of its edges have different colors.) b) Find all answersof x2 + y2 + z2 = 1 is Zp

4 a, b ∈ Z and for every n ∈ N0, the number 2na + b is a perfect square. Prove that a = 0.

5 Prove that the equationy3 = x2 + 5

doesn’t have any solutions in Z.

6 Let a, b be two positive integers and b2 + a− 1|a2 + b− 1. Prove that b2 + a− 1 has at leasttwo prime divisors.

7 Let p be a prime such that p ≡ 3 (mod 4). Prove that we can’t partition the numbersa, a + 1, a + 2, · · · , a + p− 2,(a ∈ Z) in two sets such that product of members of the sets beequal.

8 Let m,n, k be positive integers and 1 + m + n√

3 = (2 +√

3)2k+1. Prove that m is a perfectsquare.

9 Solve the equation 4xy − x− y = z2 in positive integers.

10 Let a > 1 be a positive integer. Prove that the set a2 + a− 1, a3 + a− 1, · · · have a subsetS with infinite members and for any two members of S like x, y we have gcd(x, y) = 1. Thenprove that the set of primes has infinite members.

11 Let p ≥ 3 be a prime and a1, a2, · · · , ap−2 be a sequence of positive integers such that forevery k ∈ 1, 2, · · · , p− 2 neither ak nor ak

k − 1 is divisible by p. Prove that product of someof members of this sequence is equivalent to 2 modulo p.

12 Find all subsets of N like S such that

∀m,n ∈ S =⇒ m + n

gcd(m,n)∈ S



















2007

13 Let ai∞i=1 be a sequence of positive integers such that a1 < a2 < a3 · · · and all of primesare members of this sequence. Prove that for every n < m

1an

+1

an+1+ · · ·+ 1

am6∈ N

14 Find all a, b, c ∈ N such that

a2b|a3 + b3 + c3, b2c|a3 + b3 + c3, c2a|a3 + b3 + c3.

[PS: The original problem was this: Find all a, b, c ∈ N such that

a2b|a3 + b3 + c3, b2c|a3 + b3 + c3, c2b|a3 + b3 + c3.

But I think the author meant c2a|a3 + b3 + c3, just because of symmetry]

15 Does there exists a subset of positive integers with infinite members such that for every twomembers a, b of this set

a2 − ab + b2|(ab)2

16 Prove that 22n+ 22n−1

+ 1 has at least n distinct prime divisors.

17 For a positive integer n, denote rad(n) as product of prime divisors of n. And also rad(1) = 1.Define the sequence ai∞i=1 in this way: a1 ∈ N and for every n ∈ N, an+1 = an + rad(an).Prove that for every N ∈ N, there exist N consecutive terms of this sequence which are in anarithmetic progression.

18 Prove that the equation x3 + y3 + z3 = t4 has infinitely many solutions in positive integerssuch that gcd(x, y, z, t) = 1.

19 Find all functions f : N → N such that:

i) f2000(m) = f(m).

ii) f(mn) =f(m)f(n)

f(gcd(m,n))iii) f(m) = 1 ⇐⇒ m = 1

20 Let m,n be two positive integers and m ≥ 2. We know that for every positive integer a suchthat gcd(a, n) = 1 we have n|am − 1. Prove that n ≤ 4m(2m − 1).

21 Find all primes p, q such thatpq − qp = pq2 − 19

22 Prove that for any positive integer n ≥ 3 there exist positive integers a1, a2, · · · , an such that

a1a2 · · · an ≡ ai (mod a2i ) ∀i ∈ 1, 2, · · · , n

















2007

Problem Solving Exam

1 a) Find all multiplicative functions f : Z∗p −→ Z∗

p (i.e. that ∀x, y ∈ Z∗p, f(xy) = f(x)f(y).)

b) How many bijective multiplicative does exist on Z∗p c) Let A be set of all multiplicative

functions on Z∗p, and V B be set of all bijective multiplicative functions on Z∗

p. For eachx ∈ Z∗

p, calculate the following sums :∑f∈A

f(x),∑f∈B

f(x)

2 a) Prove that center of smallest sphere containing a finite subset of Rn is inside convex hullof the point that lie on sphere. b) A is a finite subset of Rn, and distance of every two pointsof A is not larger than 1. Find radius of the largest sphere containing A.

3 Prove that for each a ∈ N, there are infinitely many natural n, such that

n | an−a+1 − 1.

4 Prove that2007∑

i=−2007

√|i + 1|

(√

2)|i|>

2007∑i=−2007

√|i|

(√

2)|i|











2008

1 Rk(m,n) is the least number such that for each coloring of k-subsets of 1, 2, . . . , Rk(m,n)with blue and red colors, there is a subset with m elements such that all of its k-subsetsare red or there is a subset with n elements such that all of its k-subsets are blue. a) If wegive a direction randomly to all edges of a graph Kn then what is the probability that theresultant graph does not have directed triangles? b) Prove that there exists a c such thatR3(4, n) ≥ 2cn.

2 Seven points are selected randomly from S1 ⊂ C. What is the probability that origin is notcontained in convex hull of these points?

3 Prove that we can put Ω(1ε ) points on surface of a sphere with radius 1 such that distance of

each of these points and the plane passing through center and two of other points is at leastε.

4 Sarah and Darah play the following game. Sarah puts n coins numbered with 1, . . . , n on atable (Each coin is in HEAD or TAIL position.) At each step Darah gives a coin to Sarahand she (Sarah) let him (Dara) to change the position of all coins with number multiple ofa desired number k. At the end, all of the coins that are in TAIL position will be given toSarah and all of the coins with HEAD position will be given to Darah. Prove that Sarah canput the coins in a position at the beginning of the game such that she gains at least Ω(n)coins. [hide=”Hint:”]Chernov inequality!

5 A permutation π is selected randomly through all n-permutations. a) if

Ca(π) = the number of cycles of length a in π

then prove that E(Ca(π)) = 1a b) Prove that if a1, a2, . . . , ak ⊂ 1, 2, . . . , n the probability

that π does not have any cycle with lengths a1, . . . , ak is at most 1Pki=1 ai











pre-preparation course exam 2004-08.pdf

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