Find all functions $f:\mathbb Z\to\mathbb Z$ such that $$19f(x)-17f(f(x))=2x$$for all $x\in\mathbb Z$.

Is it true that for any nonzero rational numbers $a$ and $b$ one can find integers $m$ and $n$ such that the number $(am+b)^2+(a+nb)^2$ is an integer? [i](M. Karpuk)[/i]

Prove that $\lfloor{\sqrt{9n+7}}\rfloor=\lfloor{\sqrt{n}+\sqrt{n+1}+\sqrt{n+2}}\rfloor$ for all postive integer $n$.

Suppose $f(x)$ is a rational function such that $3f\left(\dfrac{1}{x}\right)+\dfrac{2f(x)}{x}=x^2$ for $x\neq 0$. Find $f(-2)$.

A sequence of non-negative rational numbers $ a(1), a(2), a(3), \ldots$ satisfies $ a(m) \plus{} a(n) \equal{} a(mn)$ for arbitrary natural $ m$ and $ n$. Show that not all elements of the sequence can be distinct.

Let $m,n\in \mathbb{Z}_{\ge 1}$ and a rectangular board $m\times n$ sliced by parallel lines to the rectangle's sides into $mn$ unit squares. At moment $t=0$, there is an ant inside every square, positioned exactly in its centre, such that it is oriented towards one of the rectangle's sides. Every second, all the ants move exactly a unit following their current orientation; however, if two ants meet at the centre of a unit square, both of them turn $180^o$ around (the turn happens instantly, without any loss of time) and the next second they continue their motion following their new orientation. If two ants meet at the midpoint of a side of a unit square, they just continue moving, without changing their orientation.\\ \\ Prove that, after finitely many seconds, some ant must fall off the table.\\ \\ [i](Oliver Hayman)[/i]

Zeroes are written in all cells of a $5\times 5$ board. We can take an arbitrary cell and increase by 1 the number in this cell and the cells having a common side with it. Is it possible to obtain the number 2012 in all cells simultaneously?

In each cell of a table $8\times 8$ lives a knight or a liar. By the tradition, the knights always say the truth and the liars always lie. All the inhabitants of the table say the following statement "The number of liars in my column is (strictly) greater than the number of liars in my row". Determine how many possible configurations are compatible with the statement.

There are $4n$ pebbles of weights $1, 2, 3, \dots, 4n.$ Each pebble is coloured in one of $n$ colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied: [list] [*]The total weights of both piles are the same. [*] Each pile contains two pebbles of each colour. [/list] [i]Proposed by Milan Haiman, Hungary and Carl Schildkraut, USA[/i]

Each point on the plane is colored either red or blue. Show that there are three points of the same color that form a triangle with side lengths $1, 2,\sqrt3$.

The circumradius $R$ of a triangle with side lengths $a, b, c$ satisfies $R =\frac{a\sqrt{bc}}{b+c}$. Find the angles of the triangle.

Determine all polynomials $P(x)$ with integer coefficients which satisfies $P(n)\mid n!+2$ for all postive integer $n$.

Suppose that $y$ is a positive integer written only with digit $1$, in base $9$ system. Prove that $y$ is a triangular number, that is, exists positive integer $n$ such that the number $y$ is the sum of the $n$ natural numbers from $1$ to $n$.

Find all positive integers $n$ with the following property: there exists a polynomial $P_n(x)$ of degree $n$, with integer coefficients, such that $P_n(0)=0$ and $P_n(x)=n$ for $n$ distinct integer solutions.

Let $ABC$ be a triangle with $AB=AC$ and let $I$ be its incenter. Let $\Gamma$ be the circumcircle of $ABC$. Lines $BI$ and $CI$ intersect $\Gamma$ in two new points, $M$ and $N$ respectively. Let $D$ be another point on $\Gamma$ lying on arc $BC$ not containing $A$, and let $E,F$ be the intersections of $AD$ with $BI$ and $CI$, respectively. Let $P,Q$ be the intersections of $DM$ with $CI$ and of $DN$ with $BI$ respectively. (i) Prove that $D,I,P,Q$ lie on the same circle $\Omega$ (ii) Prove that lines $CE$ and $BF$ intersect on $\Omega$

Find all prime numbers $ p,q,r$, such that $ \frac{p}{q}\minus{}\frac{4}{r\plus{}1}\equal{}1$

Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

Place the numbers $ 1, 2, 3, 4, 5, 6, 7, 8$ at the vertices of a cuboid such that the sum of any $ 3$ numbers on a side is not less than $ 10$. Find the smallest possible sum of the 4 numbers on a side.

A coordinate system was drawn on the board and points $A (1,2)$ and $B (3,1)$ were marked. The coordinate system was erased. Restore it by the two marked points.

Let $ ABC$ be an acute triangle with the incircle $ C(I,r)$ and the circumcircle $ C(O,R)$ . Denote $ D\in BC$ for which $ AD\perp BC$ and $ AD \equal{} h_a$ . Prove that $ DI^2 \equal{} (2R \minus{} h_a)(h_a \minus{} 2r)$ .

Let $X$ be a subset of $R_{+}^{*}$ with $m$ elements. Find $X$ such that the number of subsets with the same sum is maximum.

Let $AD$ be an altitude of triangle $ABC$, and let $M$, $N$ and $P$ be midpoints of $AB$, $AD$ and $BC$, respectively. Furthermore let $K$ be a foot of perpendicular from point $D$ to line $AC$, and let $T$ be point on extension of line $KD$ (over point $D$) such that $\mid DT \mid = \mid MN \mid + \mid DK \mid$. If $\mid MP \mid = 2 \cdot \mid KN \mid$, prove that $\mid AT \mid = \mid MC \mid$.

Each of the thieves in the $n$-member party ($n \ge 3$) charged a certain number of coins. All the coins were $100n$. Thieves decided to share their prey as follows: at each step, one of the bandits puts one coin to the other two. Find them all natural numbers $n \ge 3$ for which after a finite number of steps each outlaw can have $100$ coins no matter how many coins each thug has charged.

What is largest positive integer n satisfying the following inequality: $n^{2006}$ < $7^{2007}$?

Let $n$ be an integer greater than $1$, $\alpha$ is a real, $0<\alpha < 2$, $a_1,\ldots ,a_n,c_1,\ldots ,c_n$ are all positive numbers. For $y>0$, let $$f(y)=\left(\sum_{a_i\le y} c_ia_i^2\right)^{\frac{1}{2}}+\left(\sum_{a_i>y} c_ia_i^{\alpha} \right)^{\frac{1}{\alpha}}.$$ If positive number $x$ satisfies $x\ge f(y)$ (for some $y$), prove that $f(x)\le 8^{\frac{1}{\alpha}}\cdot x$.