For any positive integer $n$, let $s(n)$ be the sum of its digits, written in decimal base. Prove that for each integer $n \ge 1$ there exists a positive integer $x$ such that the fraction $\frac{x + k}{s(x + k)}$ is not integral, for each integer $k$ with $0 \le k \le n$.

Let $ABC$ be a triangle and its incircle meets $BC, AC, AB$ at $D, E$ and $F$ respectively. Let point $ P $ on the incircle and inside $ \triangle AEF $. Let $ X=PB \cap DF , Y=PC \cap DE, Q=EX \cap FY $. Prove that the points $ A$ and $Q$ lies on $DP$ simultaneously or located opposite sides from $DP$.

$7$ pupils has been given $20$ candies, $5$ candies of $4$ different kinds, so that each pupil has no more then one candy of each kind. Prove that there are two pupils that have three or more pairs of candies of the same kind.

A polynomial $f (x)$ with real coefficients is called [i]completely reducible[/i] if it is a product of at least two non-constant polynomials whose coefficientsare all nonnegative real numbers. Show: If $f (x^{2011})$ is completely reducible, then $f(x)$ is also.

Determine all positive integers $n$ such that $$n\cdot 2^{n-1}+1$$ is a perfect square.

How many ordered pairs of sets $(A, B)$ have the properties: 1. $ A\subseteq \{1, 2, 3, 4, 5, 6\} $ 2. $ B\subseteq\{2, 3, 4, 5, 6, 7, 8\} $ 3. $ A\cap B $ has exactly $3$ elements.

Several jugs (not necessarily of the same size) with juices are placed along a circle. It is allowed to transfuse any part of juice (maybe nothing or the total content) from any jug to the neighboring one on the right, so that the latter one is not overflowed and the sugariness of its content becomes equal to $10\%$. It is known that at the initial moment such transfusion is possible from each jug. Prove that it is possible to perform several transfusions in some order, at most one transfusion from each jug, such that the sugariness of the content of each non-empty jug will become equal to $10\%$. (Sugariness is the percent of sugar in a jug, by weight. Sugar is always uniformly distributed in a jug.)

The positive real numbers $x,y,z$ satisfy $xy+yz+zx=1$. Prove that: $$ 2(x^2+y^2+z^2)+\frac{4}{3}\bigg (\frac{1}{x^2+1}+\frac{1}{y^2+1}+\frac{1}{z^2+1}\bigg) \ge 5 $$

Find all isosceles triangles that cannot be cut into three isosceles triangles with the same sides.

In a triangle $ABC$, the median $AD$ divides $\angle{BAC}$ in the ratio $1:2$. Extend $AD$ to $E$ such that $EB$ is perpendicular $AB$. Given that $BE=3,BA=4$, find the integer nearest to $BC^2$.

Michael and Andrew are playing the game Bust, which is played as follows: Michael chooses a positive integer less than or equal to $99$, and writes it on the board. Andrew then makes a move, which consists of him choosing a positive integer less than or equal to $ 8$ and increasing the integer on the board by the integer he chose. Play then alternates in this manner, with each person making exactly one move, until the integer on the board becomes greater than or equal to $100$. The person who made the last move loses. Let S be the sum of all numbers for which Michael could choose initially and win with both people playing optimally. Find S.

Prove that a necessary and sufficient for the existence of a set $ S \subset \{1,2,...,n \}$ with the property that the integers $ 0,1,...,n\minus{}1$ all have an odd number of representations in the form $ x\minus{}y, x,y \in S$, is that $ (2n\minus{}1)$ has a multiple of the form $ 2.4^k\minus{}1$ [i]L. Lovasz, J. Pelikan[/i]

Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$f(f(x)-9f(y))=(x+3y)^2f(x-3y)$$ for all $x,y\in \mathbb{R}$.

Find the set of solutions to the equation $\log_{\lfloor x\rfloor}(x^2-1)=2$

Let $S$ be a finite set of polynomials in two variables, $x$ and $y$. For $n$ a positive integer, define $ \Omega _ { n } ( S ) $ to be the collection of all expressions $ p _ { 1 } p _ { 2 } \dots p _ { k } ,$ where $p_i \in S$ and $1\leq k \leq n$. Let $d_n(S)$ indicate the maximum number of linearly independent polynomials in $ \Omega _ { n } ( S ) $. For example, $ \Omega _ { 2 } \left( \left\{ x ^ { 2 } , y \right\} \right) = \left\{ x ^ { 2 } , y , x ^ { 2 } y , x ^ { 4 } , y ^ { 2 } \right\} $ and $d _ { 2 } \left( \left\{ x ^ { 2 } , y \right\} \right) = 5 $ (a) Find $ d _ { 2 } ( \{ 1 , x , x + 1 , y \} ) $. (b) Find a closed formula in $n$ for $ d _ { n } ( \{ 1 , x , y \} ) $. (c) Calculate the least upper bound over all such sets of $ \overline{\text{lim}} _ { n \rightarrow \infty } \frac { \log d _ { n } ( S ) } { \log n } $ ($ \overline{\text{lim}} _ { n \rightarrow \infty } a _ { n } = \lim _ { n \rightarrow \infty } ( \sup \left\{ a _ { n } , a _ { n + 1 } , \ldots \right\} $, where sup means supremum or least upper bound.)

Let $a$ and $b$ be two non-zero rational numbers such that the equation $ax^2+by^2=0$ has a non-zero solution in rational numbers . Prove that for any rational number $t$ , there is a solution of the equation $ax^2+by^2=t$.

Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations: \begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*}

Show that for $n \geq 5$, the integers $1, 2, \ldots n$ can be split into two groups so that the sum of the integers in one group equals the product of the integers in the other group.

Let $ABC$ be a triangle with $AB=72,AC=98,BC=110$, and circumcircle $\Gamma$, and let $M$ be the midpoint of arc $BC$ not containing $A$ on $\Gamma$. Let $A'$ be the reflection of $A$ over $BC$, and suppose $MB$ meets $AC$ at $D$, while $MC$ meets $AB$ at $E$. If $MA'$ meets $DE$ at $F$, find the distance from $F$ to the center of $\Gamma$. [i]Proposed by Michael Kural[/i]

The value of $$\frac{\log_35\log_25}{\log_35+\log_25}$$ can be expressed as $a\log_bc$, where $a$, $b$, and $c$ are positive integers, and $a+b$ is as small as possible. Find $a+2b+3c$.

Find all functions $f : \mathbb Z \to \mathbb Z$ such that \[f (n |m|) + f (n(|m| +2)) = 2f (n(|m| +1)) \qquad \forall m,n \in \mathbb Z.\] [b]Note.[/b] $|x|$ denotes the absolute value of the integer $x.$

A scalene acute triangle has angles whose measures (in degrees) are whole numbers. What is the smallest possible measure of one of the angles, in degrees?

In the equation $ 2x^2 \minus{} hx \plus{} 2k \equal{} 0$, the sum of the roots is $ 4$ and the product of the roots is $ \minus{}3$. Then $ h$ and $ k$ have the values, respectively: $ \textbf{(A)}\ 8\text{ and }{\minus{}6} \qquad \textbf{(B)}\ 4\text{ and }{\minus{}3}\qquad \textbf{(C)}\ {\minus{}3}\text{ and }4\qquad \textbf{(D)}\ {\minus{}3}\text{ and }8\qquad \textbf{(E)}\ 8\text{ and }{\minus{}3}$

Denote by $\mathcal M_2(\mathbb R)$ the set of all $2\times2$ matrices with real entries. Prove that: a) for every $A\in\mathcal M_2(\mathbb R)$ there exist $B,C\in\mathcal M_2(\mathbb R)$ such that $A=B^2+C^2$; b) there do not exist $B,C\in\mathcal M_2(\mathbb R)$ such that $\begin{pmatrix}0&1\\1&0\end{pmatrix}=B^2+C^2$ and $BC=CB$.

Can the 'brick wall' (infinite in all directions) drawn at the picture be made of wires of length $1, 2, 3, \dots$ (each positive integral length occurs exactly once)? (Wires can be bent but should not overlap; size of a 'brick' is $1\times 2$). [asy] unitsize(0.5 cm); for(int i = 1; i <= 9; ++i) { draw((0,i)--(10,i)); } for(int i = 0; i <= 4; ++i) { for(int j = 0; j <= 4; ++j) { draw((2*i + 1,2*j)--(2*i + 1,2*j + 1)); } } for(int i = 0; i <= 3; ++i) { for(int j = 0; j <= 4; ++j) { draw((2*i + 2,2*j + 1)--(2*i + 2,2*j + 2)); } } [/asy]