Three flies are crawling along the perimeter of the triangle $ABC$ in such a way, that the centre of their masses is a constant point. One of the flies has already passed along all the perimeter. Prove that the centre of the flies' masses coincides with the centre of masses of the triangle $ABC$ . (The centre of masses for the triangle is the point of medians intersection.

Let $ABC$ be a right angled triangle with $\angle A = 90^o$and $BC = a$, $AC = b$, $AB = c$. Let $d$ be a line passing trough the incenter of triangle and intersecting the sides $AB$ and $AC$ in $P$ and $Q$, respectively. (a) Prove that $$b \cdot \left( \frac{PB}{PA}\right)+ c \cdot \left( \frac{QC}{QA}\right) =a$$ (b) Find the minimum of $$\left( \frac{PB}{PA}\right)^ 2+\left( \frac{QC}{QA}\right)^ 2$$

There are 100 visually identical coins of three types: golden, silver and copper. There is at least one coin of each type. Each golden coin weighs 3 grams, each silver coins weighs 2 grams and each copper coin weighs 1 gram. How to find the type of each coin performing no more than 101 measurements on a balance scale with no weights.

A bee is moving in three-dimensional space. A fair six-sided die with faces labeled $A^+, A^-, B^+, B^-, C^+$, and $C^-$ is rolled. Suppose the bee occupies the point $(a, b, c)$. If the die shows $A^+$, then the bee moves to the point $(a+1, b, c)$ and if the die shows $A^-$, then the bee moves to the point $(a-1, b, c)$. Analogous moves are made with the other four outcomes. Suppose the bee starts at the point $(0, 0, 0)$ and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube? $ \textbf{(A) }\frac{1}{54} \qquad \textbf{(B) }\frac{7}{54} \qquad \textbf{(C) }\frac{1}{6} \qquad \textbf{(D) }\frac{5}{18} \qquad \textbf{(E) }\frac{2}{5} \qquad $

In the coordinate plane a rectangle with vertices $ (0, 0),$ $ (m, 0),$ $ (0, n),$ $ (m, n)$ is given where both $ m$ and $ n$ are odd integers. The rectangle is partitioned into triangles in such a way that [i](i)[/i] each triangle in the partition has at least one side (to be called a “good” side) that lies on a line of the form $ x \equal{} j$ or $ y \equal{} k,$ where $ j$ and $ k$ are integers, and the altitude on this side has length 1; [i](ii)[/i] each “bad” side (i.e., a side of any triangle in the partition that is not a “good” one) is a common side of two triangles in the partition. Prove that there exist at least two triangles in the partition each of which has two good sides.

Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations: $x_{1}+2x_{2}+...+nx_{n}=0$, $x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$, ... $x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.

Find all natural numbers whose decimal notation consists of different digits of the same parity and which are perfect squares.

Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly. (a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$. (b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.

Let $m$ be a given positive integer. Prove that there exists a positive integer $k$ such that it holds $$1\leq \frac{1^m+2^m+3^m+\ldots +(k-1)^m}{k^m}<2.$$

On an infinite checkered plane $100$ chips in form of a $10\times 10$ square are given. These chips are rearranged such that any two adjacent (by side) chips are again adjacent, moreover no two chips are in the same cell. Prove that the chips are again in form of a square.

The numbers $a_1,a_2$ and $a_3$ are distinct positive integers, such that (i) $a_1$ is a divisor of $a_2+a_3+a_2a_3$; (ii) $a_2$ is a divisor of $a_3+a_1+a_3a_1$; (iii) $a_3$ is a divisor of $a_1+a_2+a_1a_2$. Prove that $a_1,a_2$ and $a_3$ cannot all be prime.

Suppose that $m$ and $n$ are relatively prime positive integers with $A = \tfrac mn$, where \[ A = \frac{2+4+6+\dots+2014}{1+3+5+\dots+2013} - \frac{1+3+5+\dots+2013}{2+4+6+\dots+2014}. \] Find $m$. In other words, find the numerator of $A$ when $A$ is written as a fraction in simplest form. [i]Proposed by Evan Chen[/i]

Given a sequence $\{a_n\}$: $a_1=1, a_{n+1}=\left\{ \begin{array}{lcr} a_n+n,\quad a_n\le n, \\ a_n-n,\quad a_n>n, \end{array} \right. \quad n=1,2,\cdots.$ Find the number of positive integers $r$ satisfying $a_r<r\le 3^{2017}$.

Consider integers $ a_i,i\equal{}\overline{1,2002}$ such that $ a_1^{ \minus{} 3} \plus{} a_2^{ \minus{} 3} \plus{} \ldots \plus{} a_{2002}^{ \minus{} 3} \equal{} \frac {1}{2}$ Prove that at least 3 of the numbers are equal.

Among $239$ coins identical in appearance there are two counterfeit coins. Both counterfeit coins have the same weight different from the weight of a genuine coin. Using a simple balance, determine in three weighings whether the counterfeit coin is heavier or lighter than the genuine coin. A simple balance shows if both sides are in equilibrium or left side is heavier or lighter. It is not required to find the counterfeit coins.

An infinite number of lilypads grow in a line, numbered $\dots$, $-2$, $-1$, $0$, $1$, $2$, $\dots$ Thumbelina and her pet frog start on one of the lilypads. She wants to make a sequence of jumps that will end on either pad $0$ or pad $96$. On each jump, Thumbelina tells her frog the distance (number of pads) to leap, but the frog chooses whether to jump left or right. From which starting pads can she always get to pad $0$ or pad $96$, regardless of her frog's decisions?

Let $k$ be a positive integer. Show that if there exists a sequence $a_0,a_1,\ldots$ of integers satisfying the condition \[a_n=\frac{a_{n-1}+n^k}{n}\text{ for all } n\geq 1,\] then $k-2$ is divisible by $3$. [i]Proposed by Okan Tekman, Turkey[/i]

Suppose real numbers $A,B,C$ such that for all real numbers $x,y,z$ the following inequality holds: \[A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0.\] Find the necessary and sufficient condition $A,B,C$ must satisfy (expressed by means of an equality or an inequality).

Let $w(x)$ be largest odd divisor of $x$. Let $a,b$ be natural numbers such that $(a,b)=1$ and \\ $a+w(b+1)$ and $b+w(a+1)$ are powers of two. Prove that $a+1$ and $b+1$ are powers of two.

It is given a tetrahedron $ABCD$ and a plane $\alpha$ intersecting the three edges passing through $D$. Prove that $\alpha$ divides the surface of the tetrahedron into two parts proportional to the volumes of the bodies formed if and only if $\alpha$ is passing through the center of the inscribed tetrahedron sphere.

The equation is given $x^2-(m+3)x+m+2=0$. If $x_1$ and $x_2$ are its solutions find all $m$ such that $\frac{x_1}{x_1+1}+\frac{x_2}{x_2+1}=\frac{13}{10}$.

For all reals $x$ define $\{x\}$ to be the difference between $x$ and the closest integer to $x$. For each positive integer $n$ evaluate : \[ S_n=\sum_{m=1}^{6n-1}\min \left(\left\{\frac{m}{6n}\right\},\left\{\frac{m}{3n}\right\}\right) \]

Let $a_0,a_1,a_2,...$ be an infinite sequence of positive integers such that $a_0 = 1$ and $a_i^2 > a_{i-1}a_{i+1}$ for all $i > 0$. (a) Prove that $a_i < a_1^i$ for all $i > 1$. (b) Prove that $a_i > i$ for all $i$.

Let $F(x)$ be a real valued function defined for all real $x$ except for $x=0$ and $x=1$ and satisfying the functional equation $F(x)+F\{(x-1)/x\}=1+x.$ Find all functions $F(x)$ satisfying these conditions.

Show that for all integer $k>1$, there are infinitely many natural numbers $n$ such that $k \cdot 2^{2^n} + 1$ is composite.