1 Given a sequence a1, a2, ... of non-negative real numbers satisfying the conditions:
1. an+ a2n≥ 3n 2. an+1+ n ≤ 2
p
an(n + 1)
for all n = 1, 2, ... indices
(1) Prove that the inequality an ≥ n holds for evere n ∈ N (2) Give an example of such a
sequence
2 Let P (x) be a polynomial with a non-negative coefficients. Prove that if the inequality P 1
x
P (x) ≥ 1 holds for x = 1, then this inequality holds for each positive x.
3 Let p, q, r be positive real numbers and n a natural number. Show that if pqr = 1, then 1 pn+ qn+ 1+ 1 qn+ rn+ 1+ 1 rn+ pn+ 1 ≤ 1.
4 Let x1, x2, ..., xn be real numbers with arithmetic mean X. Prove that there is a positive
integer K such that for any natural number i satisfying 1 ≤ i < K, we have 1 K − i
K
X
j=i+1
xj ≤
X. (In other words, the arithmetic mean of each of the lists {x1, x2, ..., xK}, {x2, x3, ..., xK},
{x3, ..., xK}, ..., {xK−1, xK}, {xK} is not greater than X.)
5 Determine the range of the following function defined for integer k, f (k) = (k)3+ (2k)5+ (3k)7− 6k
where (k)2n+1 denotes the multiple of 2n + 1 closest to k
6 A positive integer is written on each of the six faces of a cube. For each vertex of the cube we compute the product of the numbers on the three adjacent faces. The sum of these products is 1001. What is the sum of the six numbers on the faces?
7 Find all sets X consisting of at least two positive integers such that for every two elements m and n of the set X, where n ¿ m, there exists an element k of X such that n = mk2.
8 Let f (x) be a non-constant polynomial with integer coefficients, and let u be an arbitrary positive integer. Prove that there is an integer n such that f (n) has at least u distinct prime factors and f (n) 6= 0.
9 A set S of n − 1 natural numbers is given (n ≥ 3). There exist at least at least two elements in this set whose difference is not divisible by n. Prove that it is possible to choose a non-empty subset of S so that the sum of its elements is divisible by n.
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10 Is there an infinite sequence of prime numbers p1, p2, . . ., pn, pn+1, . . . such that |pn+1−2pn| =
1 for each n ∈ N?
11 Given a table m x n, in each cell of which a number +1 or -1 is written. It is known that initially exactly one -1 is in the table, all the other numbers being +1. During a move it is allowed to cell containing -1, replace this -1 by 0, and simultaneously multiply all the numbers in the neighbouring cells by -1 (we say that two cells are neighbouring if they have a common side). Find all (m,n) for which using such moves one can obtain the table containing zeros only, regardless of the cell in which the initial -1 stands.
12 There are 2n different numbers in a row. Bo one move we can onterchange any two numbers or interchange any 3 numbers cyclically (choose a, b, c and place a instead of b, b instead of c, c instead of a). What is the minimal number of moves that is always sufficient to arrange the numbers in increasing order ?
13 The 25 member states of the European Union set up a committee with the following rules: 1) the committee should meet daily; 2) at each meeting, at least one member should be represented; 3) at any two different meetings, a different set of member states should be represented; 4) at nth meeting, for every k < n, the set of states represented should include at least one state that was represented at the kth meeting.
For how many days can the committee have its meetings ?
14 We say that a pile is a set of four or more nuts. Two persons play the following game. They start with one pile of n ≥ 4 nuts. During a move a player takes one of the piles that they have and split it into two nonempty sets (these sets are not necessarily piles, they can contain arbitrary number of nuts). If the player cannot move, he loses. For which values of n does the first player have a winning strategy?
15 A circle is divided into 13 segments, numbered consecutively from 1 to 13. Five fleas called A,B,C,D and E are sitting in the segments 1,2,3,4 and 5. A flea is allowed to jump to an empty segment five positions away in either direction around the circle. Only one flea jumps at the same time, and two fleas cannot be in the same segment. After some jumps, the fleas are back in the segments 1,2,3,4,5, but possibly in some other order than they started. Which orders are possible ?
16 Through a point P exterior to a given circle pass a secant and a tangent to the circle. The secant intersects the circle at A and B, and the tangent touches the circle at C on the same side of the diameter through P as the points A and B. The projection of the point C on the diameter is called Q. Prove that the line QC bisects the angle ∠AQB.
17 Consider a rectangle with sidelengths 3 and 4, pick an arbitrary inner point on each side of this rectangle. Let x, y, z and u denote the side lengths of the quadrilateral spanned by these four points. Prove that 25 ≤ x2+ y2+ z2+ u2 ≤ 50.
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18 A ray emanating from the vertex A of the triangle ABC intersects the side BC at X and the circumcircle of triangle ABC at Y . Prove that 1
AX + 1 XY ≥
4 BC.
19 Let D be the midpoint of the side BC of a triangle ABC. Let M be a point on the side BC such that ∠BAM = ∠DAC. Further, let L be the second intersection point of the circumcircle of the triangle CAM with the side AB, and let K be the second intersection point of the circumcircle of the triangle BAM with the side AC. Prove that KL k BC. 20 Three fixed circles pass through the points A and B. Let X be a variable point on the first
circle different from A and B. The line AX intersects the other two circles at Y and Z (with Y between X and Z). Show that the ratio XY
Y Z is independent of the position of X.
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1 Let a0 be a positive integer. Define the sequence {an}n≥0 as follows: if an= j X i=0 ci10i where ci ∈ {0, 1, 2, · · · , 9}, then an+1= c20050 + c20051 + · · · + c2005j .
Is it possible to choose a0 such that all terms in the sequence are distinct?
2 Let α, β and γ be three acute angles such that sin α + sin β + sin γ = 1. Show that tan2α + tan2β + tan2γ ≥ 3
8. 3 Consider the sequence {ak}k≥1 defined by a1 = 1, a2=
1 2 and ak+2 = ak+ 1 2ak+1+ 1 4akak+1 for k ≥ 1. Prove that 1 a1a3 + 1 a2a4 + 1 a3a5 + · · · + 1 a98a100 < 4.
4 Find three different polynomials P (x) with real coefficients such that P x2+ 1 = P (x)2+ 1
for all real x.
5 Let a, b, c be positive real numbers such that abc = 1. Proove that a a2+ 2+ b b2+ 2+ c c2+ 2 ≤ 1
6 Let N and K be positive integers satisfying 1 ≤ K ≤ N . A deck of N different playing cards is shuffled by repeating the operation of reversing the order of K topmost cards and moving these to the bottom of the deck. Prove that the deck will be back in its initial order after a number of operations not greater than (2N/K)2.
7 A rectangular array has n rows and 6 columns, where n > 2. In each cell there is writ- ten either 0 or 1. All rows in the array are different from each other. For each two rows (x1, x2, x3, x4, x5, x6) and (y1, y2, y3, y4, y5, y6), the row (x1y1, x2y2, x3y3, x4y4, x5y5, x6y6) can
be found in the array as well. Prove that there is a column in which at least half of the entries are zeros.
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8 Consider a 25 × 25 grid of unit squares. Draw with a red pen contours of squares of any size on the grid. What is the minimal number of squares we must draw in order to colour all the lines of the grid?
9 A rectangle is divided into 200 × 3 unit squares. Prove that the number of ways of splitting this rectangle into rectangles of size 1 × 2 is divisible by 3.
10 Let m = 30030 and let M be the set of positive divisors of m which have exactly 2 prime factors. Determine the smallest positive integer n with the following property: for any choice of n numbers from M , there exist 3 numbers a, b, c among them satisfying abc = m.
13 What the smallest number of circles of radius √
2 that are nedeed to cover a rectangle . (a)- of size 6 ∗ 3 ? (b)- of size 5 ∗ 3 ?
16 Let n be a positive integer, let p be prime and let q be a divisor of (n + 1)p− np. Show that p divides q − 1.
19 Is it possible to find 2005 different positive square numbers such that their sum is also a square number ?
20 Find all positive integers n = p1p2· · · pk which divide (p1 + 1)(p2 + 1) · · · (pk + 1) where
p1p2· · · pk is the factorization of n into prime factors (not necessarily all distinct).
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1 Problem 1 Determine all polynomials p(x) with real coefficients such that p((x + 1)3) = (p(x) + 1)3 and p(0) = 0.
2 Problem 2 Prove that if the real numbers a, b and c satisfy a2 + b2 + c2 = 3 then X a2
2 + b + c2 ≥
(a + b + c)2
12 . When does the inequality hold?
3 Does there exist an angle α ∈ (0, π/2) such that sin α, cos α, tan α and cot α, taken in some order, are consecutive terms of an arithmetic progression?
4 The polyminal P has integer coefficients and P(x)=5 for five different integers x.Show that there is no integer x such that -7¡P(x)¡5 or 5¡P(x)¡17
5 Suppose that Romeo and Juliet each have a regular tetrahedron to the vertices of which some positive real numbers are assigned. They associate each edge of their tetrahedra with the product of the two numbers assigned to its end points. Then they write on each face of their tetrahedra the sum of the three numbers associated to its three edges. The four numbers written on the faces of Romeo’s tetrahedron turn out to coincide with the four numbers written on Juliet’s tetrahedron. Does it follow that the four numbers assigned to the vertices of Romeo’s tetrahedron are identical to the four numbers assigned to the vertices of Juliet’s tetrahedron?
6 Find all finite sets of positive integers with at least two elements such that for any two numbers a, b (a > b) belonging to the set, the number b
2
a − b belongs to the set, too. 7 How many pairs (m, n) of positive integers with m < n fulfill the equation 3
2008 = 1 m +
1 n? 8 Consider a set A of positive integers such that the least element of A equals 1001 and the
product of all elements of A is a perfect square. What is the least possible value of the greatest element of A?
9 Suppose that the positive integers a and b satisfy the equation ab− ba= 1008 Prove that a and b are congruent modulo 1008.
10 For a positive integer n, let S(n) denote the sum of its digits. Find the largest possible value of the expression S(n)
S(16n).
11 Consider a subset A of 84 elements of the set {1, 2, . . . , 169} such that no two elements in the set add up to 169. Show that A contains a perfect square.
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12 In a school class with 3n children, any two children make a common present to exactly one other child. Prove that for all odd n it is possible that the following holds: For any three children A, B and C in the class, if A and B make a present to C then A and C make a present to B.
13 For an upcoming international mathematics contest, the participating countries were asked to choose from nine combinatorics problems. Given how hard it usually is to agree, nobody was surprised that the following happened: i) Every country voted for exactly three problems. ii) Any two countries voted for different sets of problems. iii) Given any three countries, there was a problem none of them voted for. Find the maximal possible number of participating countries.
14 Is it possible to build a 4 × 4 × 4 cube from blocks of the following shape consisting of 4 unit cubes?
15 Some 1 × 2 dominoes, each covering two adjacent unit squares, are placed on a board of size n×n such that no two of them touch (not even at a corner). Given that the total area covered by the dominoes is 2008, find the least possible value of n.
16 Problem 16 Let ABCD be a parallelogram. The circle with diameter AC intersects the line BD at points P and Q. The perpendicular to the line AC passing through the point C intersects the lines AB and AD at points X and Y , respectively. Prove that the points P, Q, X and Y lie on the same circle.
Click: I proved that XYKL is cyclic (where K,L are intersection points of circle with diameter AC and AB, AD) and I tried to show that KL,XY,PQ intersect in one point but I failed... 17 Assume that a, b, c and d are the sides of a quadrilateral inscribed in a given circle. Prove
that the product (ab + cd)(ac + bd)(ad + bc) acquires its maximum when the quadrilateral is a square.
18 Let AB be a diameter of a circle S, and let L be the tangent at A. Furthermore, let c be a fixed, positive real, and consider all pairs of points X and Y lying on L, on opposite sides of A, such that |AX| · |AY | = c. The lines BX and BY intersect S at points P and Q, respectively. Show that all the lines P Q pass through a common point.
19 In a circle of diameter 1, some chords are drawn. The sum of their lengths is greater than 19. Prove that there is a diameter intersecting at least 7 chords.
20 Let M be a point on BC and N be a point on AB such that AM and CN are angle bisectors of the triangle ABC. Given that ∠BN M
∠M N C =
∠BM N
∠N M A, prove that the triangle ABC is isosceles.
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