RMO STIMULATORS << 1 >>

1<> Given a sequence un=(1+ (−1 )n )+15n+6

. Find the number of the terms of the

sequence {un } which satisfy the condition un∈( 1100 , 39100 ). [18 ]

2<< Find the sum of all irreducible fractions between 10 & 20 with denominator of 3. [300 ]

3>< There are natural numbers m∧n. Find all the fractions mn

whose

denominator is smaller than the numerator by 16, the fraction itself is smaller than the sum of the trebled inverse and 2, and the numerator is

not greater than 30. {259 , 2610

, 2711

, 2812

, 2913

, 3014 }

4>> Given : 1<a<b+c<a+1, b<c, prove that a<b .

5<< which one is larger, log3108 , log5375?

6<> Prove that 1√1

+ 1√2

+ 1√3

+…….+ 1√n

>√n for an arbitrary natural number

n≥2.

7<< Prove that |sinmα|≤m|sinα| for m∈N∧α∈R.

8>< Prove that (2n )!(n !)2

> 4 nn+1 for any natural n>1.

9>> Seven different objects must be divided among three people. In how many ways can it be done if one or two of them can get no objects.

[2187 ]

10<< How many natural numbers are there which is smaller than 104 and divisible by 4, whose decimal notations consist only of digits 0,1,2,3∧5 which do not repeat in any of these numbers?

[31 ]

11<> How many four-digit numbers are there whose decimal notation contains not more than two distinct digits?

[576 ]

12<< How many different seven-digit numbers are there the sum of whose digits is even? [45.105 ]

13>< How many different four-digit numbers can be written using the digits {1,2,3,4,5,6,7,8 } so that each of them contains only one unity, if any other

digit can occur several times in the notation of these numbers?[1372 ]

14>> How many different seven-digit numbers can be written using only three digits {1,2,3 }, under the condition that the digit 2 occurs twice in each number? [672 ]

15<< How many six-digit numbers contain exactly four different digits?[294840 ]

16<> How many different numbers, which are smaller than 2.108 and are divisible by 3, can be written by means of the digits {0,1,2 } ( numbers cannot be begin with 0)? [4373 ]

17<< How many four-digit numbers are there whose decimal notation contains not more than two different digits?

[576 ]

18>< We must form a bouquet from 18 different flowers so that it should contain not less than three flowers. How many different ways are there of forming such a bouquet? [261972 ]

19>> How many different numbers, which are smaller than 2.108, can be written by means of the digits 1∧2 ?

[766 ]

RMO STIMULATORS << 1 >>

20<< How many different six-digit numbers are there whose three digits are even and three digits are odd?

[179550 ]

21<> Prove that nn+1>(n+1 )n, n≥3 , n∈N.

22<< Brackets are to be inserted into the expression 10 @ 9 @ 8 @ 7 @ 6 @ 5 @ 4 @ 3 @ 2 so that the resulting number is an integer. ( @ means the symbol of division );

(A)Determine the maximum value of this integer.(B)Determine the minimum value of this integer.

[ 44800, 7 ]

23>< Three strictly increasing sequences

a1 , a2 , a3 ,……. ; b1 , b2 , b3 ,……. ; c1 , c2 , c3 ,……. ;

of positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer n , the following conditions hold:

(i) can=bn+1; (ii) an+1>bn (iii) the number cn+ 1cn− (n+1 ) cn+1−ncn is even.

Find a2013 ,b2011, c2010 [20132 ] ; [20112−2 ] ; [2099 ]

24>> Let a, b, c be real numbers such that for every two of the equations

x2+ax+b=0 ; x2+bx+c=0; x2+cx+a=0 ;

There is exactly one real number satisfying both of them. Determine all the possible values of a2+b2+c2 ;

25<< Find all pairs (m ,n) of integers which satisfy the equation (m+n )4=m2n2+m2+n2+6mn.

26<> Find all non-negative integer solutions of the equation 2x+2009=3y 5z.

27<< Show that an equilateral triangle is a triangle of maximum area for a given perimeter and a triangle of minimum perimeter for a given area.

28>> For any positive integer n, prove that (2n ) !<[n (n+1 ) ]n.

29<> Let the bisector of the ∠C of a ∆ ABC meet the side AB at D. Show that CD2<AC . BC;

30>> Avratanu is trying to open a lock whose code is a sequence that is three letters long, with each of the letters being one of A, B or C, possibly repeated. The lock has three buttons, labeled A, B and C. When the most recent 3 button-presses form the string, the lock opens. What is the minimum number of total button presses Avratanu needs to

be sure to open the lock?

[29]31<< Find all triplets (x; y; z) of positive integers such that

x y+ yx=z y & x y+2012= y z+1; [6, 2, 10]32<> Find all triplets (x; y; z) of real numbers such that

2 x3+1=3 zx; 2 y3+1=3 xy; 2 z3+1=3 yz; [ (1,1,1 ) ,(−12 ,−12,−12 )]

33>> Consider finitely many points in the plane with no three points on a line. All these points can be coloured red or green such that any triangle with vertices of the same colour contains at least one point of the other colour in its interior. What is the maximal possible number of points with this property? [8]

RMO STIMULATORS << 1 >>34<> The numbers from 1 to 20132 are written row by row into a table

consisting of 2013 × 2013 cells. Afterwards, all columns and all rows

containing at least one of the perfect squares 1, 4, 9,…., 20132 are simultaneously deleted. How many cells remain?

[825423]35<< The expression ±⊡±⊡±⊡±⊡±⊡±⊡ is written on the blackboard. Two

players, A and B, play a game, taking turns. Player A takes the first turn. In each turn, the player on turn replaces a symbol ⊡ by a positive integer. After all the symbols ⊡ are replaced, player A replaces each of the signs ± by either +¿ or −¿ , independently of each other. Player A wins if the value of the expressionon the blackboard is not divisible by any of the numbers 11, 12, . . . , 18.

Otherwise, player B wins.Determine which player has a winning strategy.

36>> Let f :R→R be a real function. Prove or disprove each of the following statements.

(a) If f is continuous and range (f ) = R then f is monotonic.(b) If f is monotonic and range (f ) = R then f is continuous.(c) If f is monotonic and f is continuous then range(f ) = R .

37<< Let f be a rational function (i.e. the quotient of two real polynomials) and suppose that

f (n ) is an integer for infinitely many integers n. Prove that f is a polynomial.38<> Compute the largest base-10 integer A B C D, with A > 0, such that A B C D = B! + C! + D!.

[5762]39>< Let X be the number of digits in the decimal expansion of 1001000

10000, and let Y be the number of

digits in the decimal expansion of 100010000100000. Compute [ logXY ].

[13]40<> Let α ,β be the solutions of x2−3 x+5=0. Show that for each positive

integer n, α n+βn−3n is perfectly divisible by 5.41<< Throw 6 dice at a time, find the probability, in the lowest form, such that

there will be exactly four kinds of the outcome.[0.5015 ]

42>> Let k be a constant. Find the number of the distinct positive solutions of the equationex−xe=k. [depending on k, {0,1,2,3 }]

43<> For a positive integer n, let Sn be the total sum of the intervals

of x such that sin 4 x≥ sinx in 0≤ x≤ π2

. Find limn→∞

Sn.

44<> Let a ,b be positive real numbers. Consider the circle

C1: ( x−a )2+ y2=a2 and the ellipse C2: x2+ y2

b2=1.

(1) Find the condition for which C1 is inscribed in C2.(2) Suppose b=1/√3 and C1 is inscribed in C2. Find the coordinate ( p ,q) of

the point of tangency in the first quadrant for C1 and C2.(3) Under the condition in (1), find the area of the part enclosed by C1, C2

for x≥ p.45>> Flip out three fair die at a time, find the probability such that the product of spots is divisible by 10.46<> For real number a denote by [a] the greatest integer not exceeding a.

How many positive integers n≤10000 are there which is [√n] is a divisor of n?

47>> Denote by C, L the graphs of the cubic function C : y=x3−3 x2+2 x & the line L : y=ax. (1) Find the range of a such that C and L have intersection point other

than the origin.(2) Denote S(a) by the area bounded by C and L. If a move in the range

found in (1), then find the value of a for which S(a) is minimized.48<> Let n be positive integer. Define a sequence {ak } by

a1=1

n (n+1 ) ; ak=

−1k+n+1

+ nk∑i=1

k

ai , (k=1,2,3…… )

(1) Find a2∧a3 .(2) Find the general term ak.

(3) Let bn=∑k=1

n

√ak . Prove that limn→∞

bn=ln 2.

49>> If x , y , z∈R ,x+ y+z=4 and x2+ y2+z2=6, find the maximum possible value of z .

[2]50<< Find the number of rational roots of p ( x )=2x98+3 x97+2x96+3 x95+… ..+2 x+3=0.

[2]51>< Find the least integral value of m for which every solution of the

inequality 1≤x ≤2 is a solution of x2−mx+1<0 [3]

52>> Find the number of polynomials p ( x ) with integral coefficients satisfying the conditions p (1 )=2 , p (3 )=1.

[0 ]53<< Find the degree of the remainder when x2007−1 is divided by

(x2+1 ) (x2+x+1 ).[3 ]

54>> Let p ( x ) be a polynomial with integral coefficients. Let a ,b , c be three distinct integers such that p (a )=p (b )=p (c )=−1. Find the number of integral roots of p ( x ) . [0 ]

55<> If a ,b , c>0, a2=bc and a+b+c=abc, find the least possible value of a2 . [3 ]

56>> If x1+ x2+x3=1, x1 x2+x2 x3+ x3 x1=1 and x1 x2 x3=1, find the value of |x1|+|x2|+|x3|.

[3 ]57<< find the number of real roots of p ( x )=x100−2x99+3x98−…….−100 x+101=0.

[0 ]58<> find the number of positive integral solutions of x4− y4=3879108.

[0 ]59<< Let f ( x )=a x2+2bx+c ,a ,, b , c∈R. If f ( x ) takes real values for real values of

x and imaginary values for non real x, then find the value of a.[0 ]

60>> find the real value of −a for which the roots α ,β , γ of x3−3 x2+ax−a=0 satisfy (α−3 )3+( β−3 )3+ (γ−3 )3=0. [9 ]

61<> Let a and b be two integers such that 10a+b=5 and p ( x )=x2+ax+b . Find the integer n such that p (10 ) p (11 )=p (n ).

[115 ]62>> find the number of integers solutions of x1+ x2+x3=24 subject to the

conditions 1≤x1≤5 12≤x2≤18; −1≤x3≤12. [35 ]

63<< A person starts from the origin O≡ (0,0 ) in the X-Y plane. He steps of one unit along the positive X-axis or the positive Y-axis. Travelling in this manner, find the total number of ways he can reach A≡ ( 9 ,6 ) avoiding the points P≡ (3 ,3 ) and Q≡ (6 ,4 ). [2025 ]

64<< In a collection of 1234 persons any two persons are mutually friends or enemies. Each person has at most 3 enemies. Prove that it is possible to divide this collection into two parts such that each person has at most 1 enemy in his collection.