stanford math tournament

Algebra Test2006 Stanford Math Tournament

February 25, 2006

1. A finite sequence of positive integers mi for i = 1, 2, . . . , 2006 are defined so that m1 = 1 and mi =10mi−1 + 1 for i > 1. How many of these integers are divisible by 37?

2. Find the minimum value of 2x2 + 2y2 + 5z2 − 2xy − 4yz − 4x − 2z + 15 for real numbers x, y, z.

3. A Gaussian prime is a Gaussian integer z = a + bi (where a and b are integers) with no Gaussianinteger factors of smaller absolute value. Factor −4 + 7i into Gaussian primes with positive real parts.i is a symbol with the property that i2 = −1.

4. Simplify: a3

(a−b)(a−c) + b3

(b−a)(b−c) + c3

(c−a)(c−b)

5. Jerry is bored one day, so he makes an array of Cocoa pebbles. He makes 8 equal rows with the pebblesremaining in a box. When Kramer drops by and eats one, Jerry yells at him until Kramer realizes hecan make 9 equal rows with the remaining pebbles. After Kramer eats another, he finds he can make10 equal rows with the remaining pebbles. Find the smallest number of pebbles that were in the boxin the beginning.

6. Let a, b, c be real numbers satisfying:

ab − a = b + 119bc − b = c + 59ca − c = a + 71

Determine all possible values of a + b + c.

7. Find all solutions to aabb = n4 − 6n3, where a and b are non-zero digits, and n is an integer (a and bare not necessarily distinct).

8. Evaluate:10∑

x=2

2x(x2 − 1)

.

9. Principal Skinner is thinking of two integers m and n and bets Superintendent Chalmers that he willnot be able to determine these integers with a single piece of information. Chalmers asks Skinner thenumerical value of mn+13m+13n−m2−n2. From the value of this expression alone, he miraculouslydetermines both m and n. What is the value of the above expression?

10. Evaluate:∞∑

k=1

kak−1 for |a| < 1.

1

Calculus Test2006 Stanford Math Tournament

February 25, 2006

1. Evaluate:

limx→0

ddx

sin xx

x

2. Given the equation 4y′′ + 3y′ − y = 0 and its solution y = eλt, what are the values of λ?

3. Find the volume of an hourglass constructed by revolving the graph of y = sin2(x) + 110 from −π

2 to π2

about the x-axis.

4. Evaluate

limx→0

ln(x + 1)x · ((1 + x)

12 − e)

5. Evaluate:∫

(x tan−1 x)dx

6. Evaluate ∫ π/2

0

sin3 x

sin3 x + cos3 xdx

.

7. Find Hn+1(x) in terms of Hn(x),H ′n(x),H ′′

n(x), . . . for

Hn(x) = (−1)nex2 dn

dxn

(e−x2

)8. A unicorn is tied to a cylindrical wizard’s magic tower with an elven rope stretching from the unicorn

to the top of the tower. The tower has radius 2 and height 8; the rope is of length 10. The unicornbegins as far away from the center of the tower as possible. The unicorn is startled and begins to runas close to counterclockwise as possible; as it does so the rope winds around the tower. Find the areaswept out by the shadow of the rope, assuming the sun is directly overhead. Also, you may assumethat the unicorn is a point on the ground, and that the elven rope is so light it makes a straight linefrom the unicorn to the tower.

9. Define the function tanhx = ex−e−x

ex+e−x Let tanh−1 denote the inverse function of tanh. Evaluate andsimplify:

d

dxtanh−1 tanx

10. Four ants Alan, Bill, Carl, and Diane begin at the points (0, 0), (1, 0), (1, 1), and (0, 1), respectively.Beginning at the same time they begin to walk at constant speed so that Alan is always movingdirectly toward Bill, Bill toward Carl, Carl toward Diane, and Diane toward Alan. An approximatesolution finds that after some time, Alan is at the point (0.6, 0.4). Assuming for the moment thatthis approximation is correct (it is, to better than 1%) and so the pont lies on Alan’s path, what isthe radius of curvature at that point. In standard Cartesian coordinates, the radius of curvature of afunction y(x) is given by:

R =

(1 +

(dydx

)2)3/2

∣∣∣ d2ydx2

∣∣∣

1

Geometry Test2006 Stanford Math Tournament

February 25, 2006

1. Given a cube, determine the ratio of the volume of the octahedron formed by connecting the centersof each face of the cube to the volume of the cube.

2. Given square ABCD of side length 1, with E on CD and F in the interior of the square so thatEF ⊥ DC and AF ∼= BF ∼= EF , find the area of the quadrilateral ADEF .

3. Circle γ is centered at (0, 3) with radius 1. Circle δ is externally tangent to circle γ and tangent to thex-axis. Find an equation, solved for y if possible, for the locus of possible centers (x, y) of circle δ.

4. The distance AB is l. Find the area of the locus of points X such that 15o ≤ 6 AXB ≤ 30o and X ison the same side of line AB as a given point C.

5. Let S denote a set of points (x, y, z). We define the shadow of S to be the set of points (x, y) for whichthere exists a real number z such that (x, y, z) is in S. For example, the shadow of a sphere with radiusr centered on the z axis is a circle in the xy plane centered at the origin with radius r. Suppose a cubehas a shadow consisting of a regular hexagon witih area 147

√3. What is the side length of the cube?

6. A circle of radius R is placed tangent to two perpendicular lines. Another circle is placed tangent tothe same two lines and the first circle. In terms of R, what is the radius of a third circle that is tangentto one line and tangent to both other circles?

7. A certain 2’ by 1’ pool table has pockets, denoted [A, . . . , F ] as shown. A pool player strikes a ballat point x, 1

4 of the way up side AC, aiming for a point 1.6’ up the opposite side of the table. Hemakes his mark, and the ball ricochets around the edges of the table until it finally lands in one of thepockets. How many times does it ricochet before it falls into a pocket, and which pocket? Write youranswer in the form {C, 2006}.

8. In triangle 4PQR, the altitudes from P,Q and R measure 5, 4 and 4, respectively. Find QR2.

9. Poles A,B, and P1, P2, P3,. . . are vertical line segments with bases on the x-axis. The tops of poles Aand B are (0,1) and (200,5), respectively. A string S connects (0,1) and (200,0) and intersects anotherstring connecting (0,0) and (200,5) at point T. Pole P1 is constructed with T as its top point. For eachinteger i > 1, pole Pi is constructed so that its top point is the intersection of S and the line segmentconnecting the base of Pi−1 (on the x-axis) and the top of pole B. Find the height of pole P100.

10. In triangle 4ABC, points P,Q and R lie on sides AB,BC and AC, respectively, so that APPB = BQ

QC =CRRA = 1

3 . If the area of 4ABC is 1, determine the area of the triangle formed by the points ofintersection of lines AQ,BR and CP .

1

Algebra Test2007 Stanford Math Tournament

March 4, 2007

1. Find all real roots of f if f(x1/9) = x2 − 3x− 4.

2. Given that x1 > 0 and x2 = 4x1 are solutions to ax2 + bx + c and that 3a = 2(c− b), what is x1?

3. Let a, b, c be the roots of x3 − 7x2 − 6x + 5 = 0. Compute (a + b)(a + c)(b + c).

4. How many positive integers n, with n ≤ 2007, yield a solution for x (where x is real) in the equationbxc+ b2xc+ b3xc = n?

5. The polynomial −400x5 + 2660x4 − 3602x3 + 1510x2 + 18x− 90 has five rational roots. Suppose youguess a rational number which could possibly be a root (according to the rational root theorem). Whatis the probability that it actually is a root?

6. What is the largest prime factor of 49 + 94?

7. Find the minimum value of xy + x + y + 1xy + 1

x + 1y for x, y > 0 real.

8. If r + s + t = 3, r2 + s2 + t2 = 1, and r3 + s3 + t3 = 3, compute rst.

9. Find a2 + b2 given that a, b are real and satisfy

a = b +1

a + 1b+ 1

a+···

; b = a− 1b + 1

a− 1b+···

10. Evaluate2007∑k=1

(−1)kk2

Calculus Test2007 Stanford Math Tournament

March 4, 2007

1. Findlimx→0

−1 + cos x

3x2 + 4x3

2. A line through the origin is tangent to y = x3 + 3x + 1 at the point (a, b). What is a?

3. A boat springs a leak at time t = 0, with water coming in at constant rate. At a time t = τ > 0 hours,someone notices that there is a leak and starts to record distance the boat travels. The boat’s speed isinversely related to the amount of water in the boat. If the boat travels twice as far in the first houras in the second hour, what is τ?

4. Let I(n) =∫ π

0sin(nx)dx. Find

∞∑n=0

I(5n).

5. Let Θk(x) be 0 for x < k and 1 for x ≥ k. The Dirac delta “function” is defined to be δk(x) = ddxΘk(x).

(It’s really called a distribution, and we promise it makes sense.) Suppose d2

dx2 f(x) = δ1(x)+ δ2(x) andf(0) = f ′(0) = 0. What is f(5)?

6. Point A is chosen randomly from the circumference of the unit circle, while point B is chosen randomlyin the interior. A rectangle is then constructed using A,B as opposite vertices, with sides parallel orperpendicular to the coordinate axes. What is the probability that the rectangle lies entirely insidethe circle?

7. A balloon in cross-section has the equation y = ±√

2x− x2e−x/2, with the x-axis beginning at the topof the balloon pointing toward the knot at the bottom. What is its volume?

8. Silas does nothing but sleep, drink coffee, and prove theorems, and he never more than one at a time.It takes 5 minutes to drink a cup of coffee. When doing math, Silas proves s + ln c theorems per hour,where c is the number of cups of coffee he drinks per day, and s is the number of hours he sleeps perday. How much coffee should Silas get in a day to prove the most theorems?

9. Evaluate limn→∞

2n∑k=n+1

1k .

10. Find the 10th nonzero term of the power series for f(x) = x(x2−1)2 (expanding about x = 0).

Geometry Test2007 Stanford Math Tournament

March 4, 2007

1. An equilateral triangle has perimeter numerically equal to its area, which is not zero. Find its sidelength.

2. Two spheres of radius 2 pass through each other’s center. Find the surface area of the regular octahe-dron inscribed within the space enclosed by both spheres.

3. Cumulation of a polyhedron means replacing each face with a pyramid of height h using the face as abase. There is a cumulation of the cube of side length s which (after removing unecessary edges) hastwelve sides, each a congruent rhombus. What is the height h used in this cumulation?

4. Nathan is standing on vertex A of triangle ABC, with AB = 3, BC = 5, and CA = 4. Nathan walksaccording to the following plan: He moves along the altitude-to-the-hypotenuse until he reaches thehypotenuse. He has now cut the original triangle into two triangles; he now walks along the altitudeto the hypotenuse of the larger one. He repeats this process forever. What is the total distance thatNathan walks?

5. Given an octahedron with every edge of length s, what is the radius of the largest sphere that will fitin this octahedron?

6. Let TINA be a quadrilateral with IA = 8, IN = 4, m 6 T = 30◦, m 6 NAT = 60◦, and m6 TIA =m6 INA. Find NA.

7. Two regular tetrahedra of side length 2 are positioned such that the midpoint of each side of onecoincides with the midpoint of a side of the other, and the tetrahedra themselves do not coincide. Findthe volume of the region in which they overlap.

8. ∆ABC has AB = AC. Points M and N are midpoints of AB and AC, respectively. The mediansMC and NB intersect at a right angle. Find (AB

BC)2.

9. Points P,Q,R, S, T lie in the pane with S on PR and R on QT . If PQ = 5, PS = 3, PR = 5, QS = 3,and RT = 4/

√3, what is ST?

10. A car starts moving at constant speed at the origin facing in the positive y-direction. Its minimumturning radius is such that it the soonest it can return to the x-axis is after driving a distance d. LetΓ be the boundary of the region the car can reach by driving at most a distance d; find an x > 0 sothat

(x, d

3 + d√

32π

)is on Γ.

SMT 2008 Algebra Test February 23, 2008

1. Reid is twice as old as Gabe. Four years ago, Gabe was twice as old as Dani. In 10 years, Reid will betwice as old as Dani. How many years old is Reid now?

2. Let P (x) = x6 + ax5 + bx4 + x3 + bx2 + ax + 1. Given that 1 is a root of P (x) = 0 and −1 is not, whatis the maximum number of distinct real roots that P could have?

3. If a, b, c ∈ C and a + b + c = ab + bc + ac = abc = 1, find a, b, c. (The order in which you write youranswers does not matter.)

4. Find x4 + y4 + z4, given that {0 = x + y + z

1 = x2 + yz + z2

5. The product of a 13x5 matrix and a 5x13 matrix contains the entry x in exactly two places. If D(x) isthe determinant of the matrix product, D(x = 0) = 2008, D(x = −1) = 1950, and D(x = 2) = 2142.Find D(x).

6. For how many integers k, with 0 ≤ k ≤ 2008, does x2 − x− k = 0 have integer solutions for x?

7. Find all ordered pairs of positive integers (p, q) such that 2p2 + q2 = 4608.

8. How many monic polynomials P (x) are there with P (x)Q(x) = x4−1 for some other polynomial Q(x),where the coefficients of P and Q are in C?

9. Find the number of distinct ordered integer pairs (x,y) with x + y − xy = 43.

10. Evaluate∞∑

k=1

k

5k.

SMT 2008 Calculus Test February 23, 2008

1. Compute∫ π/2

0

sin x cos x dx.

2. Evaluate:

limx→0

10x2 − 12x3

e13x

2 − 1

3. Find the area enclosed by the graph given by the parametric equations

y = sin(2t)x = sin(t)

4. Find the value of the nth derivative of f(x) = sinn(x) at x = 0.

5. Water flows into a tank at 3 gallons per minute. The tank initially contains 100 gallons of water, with50 pounds of salt. The tank is well-mixed, and drains at a rate of 2 gallons per minute. How manypounds of salt are left after one hour?

6. Evaluate∫

e3x sin(x)dx.

7. Compute∞∑n=0

2n−1

n!.

8. Find f(x) such that limh→0

h2

f(x+2x)−2f(x+h)+f(x) = −x3

2 − x− 12x .

9. Suppose x′′(t) + x′(t) = t5x(t). Let the power series representation of x be x(t) =∑

antn. Find an in

terms of an−1 and an−7, where n > 7.

10. Evaluate: ∫ x

−∞t2tetdt

SMT 2008 Geometry Test February 23, 2008

1. A regular polygon of side length 1 has the property that if regular pentagons of side length 1 are placedon each side, then each pentagon shares a side with the two adjacent ones. How many sides does sucha polygon have?

2. John stands against one wall of a square room with walls of length 4 meters each. He kicks a frictionless,perfectly elastic ball in such a way that it bounces off the three other walls once each and returns tohim (diagram not geometrically accurate). How many meters does the ball travel?

3. A cube is inscribed in a sphere of radius r. Find the ratio of the volume of the cube to that of thesphere.

4. A circle of radius 144 has three smaller circles inside it, all congruent. Each small circle is tangent tothe other two and to the large circle. Find the radius of one of the smaller circles.

5. In 4ABC, ∠C is right, AC = 2−√

3 + x and BC = 1− 2x + x√

3. Find m∠B.

6. Points A, B,C lie on sides DE, EF , and FE of 4DEF , respectively. If DA = 3, AE = 2, EB = 2,BF = 11, FC = 11, and CD = 1, find the area of 4ABC.

7. What is the area of the incircle of a triangle with side lengths 10040, 6024, and 8032?

8. Rhombus ABCD has side length l, with cos(m∠B) = − 23 . The circle through points A, B, and D has

radius 1. Find l.

9. A trapezoid has bases of length 10 and 15. Find the length of the segment that stretches from one legof the trapezoid to the other, parallel to the bases, through the intersection point of the diagonals.

10. A regular polygon with 40 sides, all of length 1, is divided into triangles, with each vertex of eachtriangle being a vertex of the original polygon. Let A be the area of the smallest triangle. What is theminimum number of square root signs needed to express the exact value of A?


1. No math tournament exam is complete without a self referencing question. What is the product ofthe smallest prime factor of the number of words in this problem times the largest prime factor of thenumber of words in this problem?

2. King Midas spent 100x % of his gold deposit yesterday. He is set to earn gold today. What percentage

of the amount of gold King Midas currently has would he need to earn today to end up with as muchgold as he started?

3. Find all integer pairs (a, b) such that ab+ a− 3b = 5.

4. Find all values of x for which f(x) + xf(

1x

)= x for any function f(x).

5. Find the minimum possible value of 2x2 + 2xy + 4y + 5y2 − x for real numbers x and y.

6. The dollar is now worth 1980 ounce of gold. After the nth $7001 billion “No Bank Left Behind” bailout

package passed by congress, the dollar gains 1

22n−1 of its (n − 1)th value in gold. After four bankbailouts, the dollar is worth 1

b

(1− 1

2c

)in gold, where b,c are positive integers. Find b+ c.

7. Evaluate2009∑k=1

b k60c

8. “Balanced tertiary” is a positional notation system in which numbers are written in terms of the digits1̄ (negative one), 0, and 1 with the base 3. For instance, 101̄1 = (1)30 + (−1)31 + (0)32 + 1(3)3 = 2510.Calculate (11̄00)(1̄1) + (11̄1) and express your answer in balanced tertiary.

9. All the roots of x3 + ax2 + bx + c are positive integers greater than 2, and the coefficients satisfya+ b+ c+ 1 = −2009. Find a.

10. Let δ(n) be the number of 1s in the binary expansion of n (e.g. δ(1) = 1, δ(2) = 1, δ(3) = 2, δ(4) = 1).Evaluate:

10

( ∑∞n=1

δ(n)n2∑∞

n=0(−1)n−1δ(n)

n2

).


1. Find the exact value of 1− 13!

+15!− . . ..

2. At SMT 2008, we met a man named Bill who has an infinite amount of time. This year, he is walkingcontinuously at a speed of 1

1+t2 , starting at time t = 0. If he continues to walk for an infinite amountof time, how far will he walk?

3. Evaluate limx→0

10x2

sin2(3x).

4. Compute∫ 1

0

tan−1(x)dx

5. Let a(t) = cos2(2t) be the acceleration at time t of a point particle traveling on a straight line. Supposeat time t = 0, the particle is at position x = 1 with velocity v = −2. Find its position at time t = 2.

6. Find∞∑

n=2

dn

dxn(e−ax)

for |a| < 1.

7. Compute

limn→∞

n∑k=1

n− k

n2cos(

4k

n

).

8. Evaluate∫∞0

4bx + 7ce−2xdx. Remember to express your answer as a single fraction.

9. Compute∞∑

n=0

n

(15

)n

.

10. Evaluate∞∑

n=1

150 + n2/80000

, as a decimal to the nearest tenth.


1. The sum of all of the interior angles of seven polygons is 180 · 17. Find the total number of sides of thepolygons.

2. The pattern in the figure below continues inward infinitely. The base of the biggest triangle is 1. Alltriangles are equilateral. Find the shaded area.

3. Given a regular pentagon, find the ratio of its diagonal, d, to its side, a.

4. ABCD form a rhobus. E is the intersection of AC and BD. F lie on AD such that EF ⊥ FD. GivenEF = 2 and FD = 1. Find the area of the rhobus ABCD.

5. In the 2009 Stanford Olympics, Willy and Sammy are two bikers. The circular race track has twolanes, the inner lane with radius 11, and the outer with radius 12. Willy will start on the inner lane,and Sammy on the outer. They will race for one complete lap, measured by the inner track. What isthe square of the distance between Willy and Sammy’s starting positions so that they will both racethe same distance? Assume that they are of point size and ride perfectly along their respective lanes.

6. Equilateral triangle ABC has side length of 24. Points D, E, F lie on sides BC, CA, AB such thatAD ⊥ BC, DE ⊥ AC, and EF ⊥ AB. G is the intersection of AD and EF . Find the area of thequadrilateral BFGD.

7. Four disks with disjoint interiors are mutually tangent. Three of them are equal in size and the fourthone is smaller. Find the ratio of the radius of the smaller disk to one of the larger disks.

8. Three points are randomly placed on a circle. What is the probability that they lie on the samesemicircle?

9. Two circles with centers A and B intersect at points X and Y . The minor arc ∠XY = 120◦ withrespect to circle A, and ∠XY = 60◦ with respect to circle B. If XY = 2, find the area shared by thetwo circles.

10. Right triangle ABC is inscribed in circle W . ∠CAB = 65◦, and ∠CBA = 25◦. The median from Cto AB intersects W at D. Line l1 is drawn tangent to W at A. Line l2 is drawn tangent to W at D.The lines l1 and l2 intersect at P . Compute ∠APD.


1. Compute

√√√√1 +

√1 +

√1 +

√1 +

√1 +√

1 + ....

2. Write 0.2010228 as a fraction.

3. Bob sends a secret message to Alice using her RSA public key n = 400000001. Eve wants to listen inon their conversation. But to do this, she needs Alice’s private key, which is the factorization of n.Eve knows that n = pq, a product of two prime factors. Find p and q.

4. If x2 + 1/x2 = 7, find all possible values of x5 + 1/x5.

5. A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opensevery locker. Student 3 goes through and “flips” every 3rd locker (“flipping”) a locker means changingits state: if the locker is open he closes it, and if the locker is closed he opens it. Student 5 thengoes through and “flips” every 5th locker. This process continues with all students with odd numbersn < 100 going through and “flipping” every nth locker. How many lockers are open after this process?

6. Consider the sequence 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, ... Find n such that the first n terms sum upto 2010.

7. Find all the integers x in [20, 50] such that 6x + 5 ≡ −19 mod 10, that is, 10 divides (6x + 15) + 19.

8. Let P (x) be a polynomial of degree n such that P (k) = 3k for 0 ≤ k ≤ n. Find P (n + 1).

9. Suppose xy− 5x + 2y = 30, where x and y are positive integers. Find the sum of all possible values ofx.

10. Find the sum of all solutions of the equation

1x2 − 1

+2

x2 − 2+

3x2 − 3

+4

x2 − 4= 2010x− 4.


1. Evaluate

limt→0

1t

(tan−1

(1

x + t

)− tan−1

(1x

))2. Find the minimum value of ex − x− x3

3 .

3. Given∫ ∞−∞

etxf(x) dx = sin−1(t−√

1/2)

, find∫ ∞−∞

xf(x) dx.

4. Find the values of x that maximize f(x) =∣∣∣∣ 3x + 19x2 + 6x + 2

∣∣∣∣.5. A rectangular pyramid tower is being built on a circular island of radius two. The height of the tower

is equal to its width. What is the maximum volume of the tower?

6. Evaluate∞∑

k=0

ke−13 13k

k!.

7. Calculated

dt

[∫ ln 1/t

− ln 1/t

cos(tex)dx

].

8. Evaluate11

+12− 2

3+

14

+15− 2

6+

17

+18− 2

9+ · · ·

9. Find the value of k which minimizes

F (k) =∫ 4

0

|x(4− x)− k| dx.

10. Let f(x) = x6 − 6x2 + 6x − 7. It is known that this polynomial has three critical points. Find theparabola passing through these critical points.


Note: Figures may not be drawn to scale.

1. Find the reflection of the point (11, 16, 22) across the plane 3x + 4y + 5z = 7.

2. Find the radius of a circle inscribed in a triangle with side lengths 4, 5, and 6.

3. Find the volume of a regular cubeoctahedron of side length 1. This is a solid whose faces comprise 6squares and 8 equilateral triangles, arranged as in the diagram below.

4. Given triangle ABC. D lies on BC such that AD bisects ∠BAC. Given AB = 3, AC = 9, andBC = 8. Find AD.

5. Find the sum of angles A, B,C, D, E, F,G, H, I in the following diagram:A

D

G

I

B

E

H

C

F

6. In the diagram below, let OT = 25 and AM = MB = 30. Find MD.


T

D

M

B

A

O

7. 4ABC is a triangle with AB = 5, BC = 6, and CA = 7. Squares are drawn on each side, as in theimage below. Find the area of hexagon DEFGHI.

H

I

A D

H

A D

G

C

G

B

EF

8. A sphere of radius 1 is internally tangent to all four faces of a regular tetrahedron. Find the tetrahe-dron’s volume.

9. For an acute triangle 4ABC and a point X satisfying ∠ABX + ∠ACX = ∠CBX + ∠BCX, find theminimum length of AX if AB = 13, BC = 14, and CA = 15.

10. A, B, C, D are points along a circle, in that order. AC intersects BD at X. If BC = 6, BX = 4,XD = 5, and AC = 11, find AB.


Time limit: 50 minutes.Instructions: This test contains 10 short answer questions. All answers must be expressed in simplest formunless specified otherwise. Only answers written on the answer sheet will be considered for grading.No calculators.

1. Let a, b ∈ C such that a + b = a2 + b2 = 2√3

3 i. Compute |Re(a)|.

2. Consider the curves x2 + y2 = 1 and 2x2 + 2xy + y2 − 2x − 2y = 0. These curves intersect at twopoints, one of which is (1, 0). Find the other one.

3. If r, s, t, and u denote the roots of the polynomial f(x) = x4 + 3x3 + 3x + 2, find

1

r2+

1

s2+

1

t2+

1

u2.

4. Find the 2011th-smallest x, with x > 1, that satisfies the following relation:

sin(lnx) + 2 cos(3 lnx) sin(2 lnx) = 0.

5. Find the remainder when (x + 2)2011 − (x + 1)2011 is divided by x2 + x + 1.

6. There are 2011 positive numbers with both their sum and the sum of their reciprocals equal to 2012.Let x be one of these numbers. Find the maximum of x + x−1.

7. Let P (x) be a polynomial of degree 2011 such that P (1) = 0, P (2) = 1, P (4) = 2, ... , and P (22011) =2011. Compute the coefficient of the x1 term in P (x).

8. Find the maximum ofab + bc + cd

a2 + b2 + c2 + d2

for reals a, b, c, and d not all zero.

9. It is a well-known fact that the sum of the first n k-th powers can be represented as a polynomial inn. Let Pk(n) be such a polynomial for integers k and n. For example,

n∑i=1

i2 =n(n + 1)(2n + 1)

6,

so one has

P2(x) =x(x + 1)(2x + 1)

6=

1

3x3 +

1

2x2 +

1

6x.

Evaluate P7(−3) + P6(−4).

10. How many polynomials P of degree 4 satisfy P (x2) = P (x)P (−x)?



1. If f(x) = (x− 1)4(x− 2)3(x− 3)2, find f ′′′(1) + f ′′(2) + f ′(3).

2. A trapezoid is inscribed in a semicircle of radius 2 such that one base of the trapezoid lies along thediameter of the semicircle. Find the largest possible area of the trapezoid.

3. A sector of a circle has angle θ. Find the value of θ, in radians, for which the ratio of the sector’s areato the square of its perimeter (the arc along the circle and the two radial edges) is maximized. Expressyour answer as a number between 0 and 2π.

4. Let f(x) = x3ex2

1−x2 . Find f (7)(0), the 7th derivative of f evaluated at 0.

5. The real-valued infinitely differentiable function f(x) is such that f(0) = 1, f ′(0) = 2, and f ′′(0) = 3.Furthermore, f has the property that

f (n)(x) + f (n+1)(x) + f (n+2)(x) + f (n+3)(x) = 0

for all n ≥ 0, where f (n)(x) denotes the nth derivative of f . Find f(x).

6. Compute

∫ π

−π

x2

1 + sinx+√

1 + sin2 xdx.

7. For the curve sin(x) + sin(y) = 1 lying in the first quadrant, find the constant α such that

limx→0

xαd2y

dx2

exists and is nonzero.

8. Compute

∫ 2

12

tan−1 x

x2 − x+ 1dx.

9. Solve the integral equation

f(x) =

∫ x

0

ex−yf ′(y) dy − (x2 − x+ 1)ex.

10. Compute the integral ∫ π

0

ln(1 − 2a cosx+ a2) dx

for a > 1.



1. Triangle ABC has side lengths BC = 3, AC = 4, AB = 5. Let P be a point inside or on triangle ABCand let the lengths of the perpendiculars from P to BC,AC,AB be Da, Db, Dc respectively. Computethe minimum of Da +Db +Dc.

2. Pentagon ABCDE is inscribed in a circle of radius 1. If ∠DEA ∼= ∠EAB ∼= ∠ABC, m∠CAD = 60◦,and BC = 2DE, compute the area of ABCDE.

3. Let circle O have radius 5 with diameter AE. Point F is outside circle O such that lines FA and FEintersect circle O at points B and D, respectively. If FA = 10 and m∠FAE = 30◦, then the perimeterof quadrilateral ABDE can be expressed as a+ b

√2 + c

√3 + d

√6, where a, b, c, and d are rational.

Find a+ b+ c+ d.

4. Let ABC be any triangle, and D,E, F be points on BC, CA, AB such that CD = 2BD, AE = 2CEand BF = 2AF . AD and BE intersect at X, BE and CF intersect at Y , and CF and AD intersect

at Z. Find Area(4ABC)Area(4XY Z) .

5. Let ABCD be a cyclic quadrilateral with AB = 6, BC = 12, CD = 3, and DA = 6. Let E,F be theintersection of lines AB and CD, lines AD and BC respectively. Find EF .

6. Two parallel lines l1 and l2 lie on a plane, distance d apart. On l1 there are an infinite number ofpoints A1, A2, A3, · · · , in that order, with AnAn+1 = 2 for all n. On l2 there are an infinite number ofpoints B1, B2, B3, · · · , in that order and in the same direction, satisfying BnBn+1 = 1 for all n. Giventhat A1B1 is perpendicular to both l1 and l2, express the sum

∑∞i=1 ∠AiBiAi+1 in terms of d.

7. In a unit square ABCD, find the minimum of√

2AP +BP + CP where P is a point inside ABCD.

8. We have a unit cube ABCDEFGH where ABCD is the top side and EFGH is the bottom side withE below A, F below B, and so on. Equilateral triangle BDG cuts out a circle from the cube’s inscribedsphere. Find the area of the circle.

9. We have a circle O with radius 10 and four smaller circles O1, O2, O3, O4 of radius 1 which are internallytangent toO, with their tangent points toO in counterclockwise order. The small circles do not intersecteach other. Among the two common external tangents of O1 and O2, let l12 be the one which separatesO1 and O2 from the other two circles, and let the intersections of l12 and O be A1 and B2, with A1

denoting the point closer to O1. Define l23, l34, l41 and A2, A3, A4, B3, B4, B1 similarly. Suppose thatthe arcs A1B1, A2B2, and A3B3 have length π, 3π/2, and 5π/2 respectively. Find the arc length ofA4B4.

10. Given a triangle ABC with BC = 5, AC = 7, and AB = 8, find the side length of the largest equilateraltriangle PQR such that A,B,C lie on QR,RP, PQ respectively.


Time limit: 50 minutes.Instructions: This test contains 10 short answer questions. All answers must be expressed in sim-plest form unless specified otherwise. Only answers written on the answer sheet will be consideredfor grading.No calculators.

1. Compute the minimum possible value of

(x − 1)2 + (x − 2)2 + (x − 3)2 + (x − 4)2 + (x − 5)2

for real values of x.

2. Find all real values of x such that (15(x2 − 10x + 26))x2−6x+5 = 1.

3. Express 23−123+1

× 33−133+1

× 43−143+1

× · · · × 163−1163+1

as a fraction in lowest terms.

4. If x, y, and z are integers satisfying xyz +4(x+ y + z) = 2(xy +xz + yz)+7, list all possibilitiesfor the ordered triple (x, y, z).

5. The quartic (4th-degree) polynomial P (x) satisfies P (1) = 0 and attains its maximum value of3 at both x = 2 and x = 3. Compute P (5).

6. There exist two triples of real numbers (a, b, c) such that a − 1b , b − 1

c , and c − 1a are the roots

to the cubic equation x3 − 5x2 − 15x + 3 listed in increasing order. Denote those (a1, b1, c1) and(a2, b2, c2). If a1, b1, and c1 are the roots to monic cubic polynomial f and a2, b2, and c2 are theroots to monic cubic polynomial g, find f(0)3 + g(0)3.

7. The function f(x) is known to be of the form∏n

i=1 fi(aix), where ai is a real number and fi(x)is either sin(x) or cos(x) for i = 1, . . . , n. Additionally, f(x) is known to have zeros at everyinteger between 1 and 2012 (inclusive) except for one integer b. Find the sum of all possiblevalues of b.

8. For real numbers (x, y, z) satisfying the following equations, find all possible values of x + y + z.

x2y + y2z + z2x = −1

xy2 + yz2 + zx2 = 5xyz = −2

9. Find the minimum value of xy, given that x2 + y2 + z2 = 7, xy + xz + yz = 4, and x, y, z arereal numbers.

10. Let X1, X2, . . . , X2012 be chosen independently and uniformly at random from the interval (0, 1].In other words, for each Xn, the probability that it is in the interval (a, b] is b − a. Computethe probability that dlog2 X1e + dlog4 X2e + · · · + dlog4024 X2012e is even. (Note: For any realnumber a, dae is defined as the smallest integer not less than a.)



1. What is∫ 100 (x− 5) + (x− 5)2 + (x− 5)3 dx?

2. Find the maximum value of ∫ 3π/2

−π/2sin(x)f(x) dx

subject to the constraint |f(x)| ≤ 5.

3. Calculate ∫ 35

25

1x− x3/5

dx.

4. Compute the x-coordinate of the point on the curve y =√

x that is closest to the point (2, 1).

5. Let

f(x) = x +x2

2+

x3

3+

x4

4+

x5

5,

and set g(x) = f−1(x). Compute g(3)(0).

6. Compute

limx→0

(sinx

x

) 11−cos x

.

7. A differentiable function g satisfies∫ x

0(x− t + 1)g(t) dt = x4 + x2

for all x ≥ 0. Find g(x).

8. Compute ∫ ∞

0

lnx

x2 + 4dx.

9. Find the ordered pair (α, β) with non-infinite β 6= 0 such that limn→∞

n2√1!2! · · ·n!

nα= β holds.

10. Find the maximum of ∫ 1

0f(x)3 dx

given the constraints

−1 ≤ f(x) ≤ 1,

∫ 1

0f(x) dx = 0.



1. A circle with radius 1 has diameter AB. C lies on this circle such that_

AC /_

BC= 4. AC dividesthe circle into two parts, and we will label the smaller part Region I. Similarly, BC also dividesthe circle into two parts, and we will denote the smaller one as Region II. Find the positivedifference between the areas of Regions I and II.

2. In trapezoid ABCD, BC ‖ AD, AB = 13, BC = 15, CD = 14, and DA = 30. Find the area ofABCD.

3. Let ABC be an equilateral triangle with side length 1. Draw three circles Oa, Ob, and Oc withdiameters BC, CA, and AB, respectively. Let Sa denote the area of the region inside Oa andoutside of Ob and Oc. Define Sb and Sc similarly, and let S be the area of the region inside allthree circles. Find Sa + Sb + Sc − S.

4. Let ABCD be a rectangle with area 2012. There exist points E on AB and F on CD suchthat DE = EF = FB. Diagonal AC intersects DE at X and EF at Y . Compute the area oftriangle EXY .

5. What is the radius of the largest sphere that fits inside an octahedron of side length 1?

6. A red unit cube ABCDEFGH (with E below A, F below B, etc.) is pushed into the cornerof a room with vertex E not visible, so that faces ABFE and ADHE are adjacent to the walland face EFGH is adjacent to the floor. A string of length 2 is dipped in black paint, and oneof its endpoints is attached to vertex A. How much surface area on the three visible faces of thecube can be painted black by sweeping the string over it?

7. Let ABC be a triangle with incircle O and side lengths 5, 8, and 9. Consider the other tangentline to O parallel to BC, which intersects AB at Ba and AC at Ca. Let ra be the inradius oftriangle ABaCa, and define rb and rc similarly. Find ra + rb + rc.

8. Let ABC be a triangle with side lengths 5, 6, and 7. Choose a radius r and three points outsidethe triangle Oa, Ob, and Oc, and draw three circles with radius r centered at these three points.If circles Oa and Ob intersect at C, Ob and Oc intersect at A, Oc and Oa intersect at B, and allthree circles intersect at a fourth point, find r.

9. In quadrilateral ABCD, m∠ABD ∼= m∠BCD and ∠ADB = ∠ABD + ∠BDC. If AB = 8 andAD = 5, find BC.

10. A large flat plate of glass is suspended√

2/3 units above a large flat plate of wood. (The glassis infinitely thin and causes no funny refractive effects.) A point source of light is suspended

√6

units above the glass plate. An object rests on the glass plate of the following description. Itsbase is an isosceles trapezoid ABCD with AB‖DC, AB = AD = BC = 1, and DC = 2. Thepoint source of light is directly above the midpoint of CD. The object’s upper face is a triangleEFG with EF = 2, EG = FG =

√3. G and AB lie on opposite sides of the rectangle EFCD.

The other sides of the object are EA = ED = 1, FB = FC = 1, and GD = GC = 2. Computethe area of the shadow that the object casts on the wood plate.

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