Does there exist a subset $E$ of the set $N$ of all positive integers such that none of the elements in $E$ can be presented as a sum of at least two other (not necessarily distinct) elements from $E$ ? (E. Barabanov)

Points $H$ and $T$ are marked respectively on the sides $BC$ abd $AC$ of triangle $ABC$ so that $AH$ is the altitude and $BT$ is the bisectrix $ABC$. It is known that the gravity center of $ABC$ lies on the line $HT$. a) Find $AC$ if $BC$=a and $AB$=c. b) Determine all possible values of $\frac{c}{a}$ for all triangles $ABC$ satisfying the given condition. (I. Voronovich) reason for this postposted for the related contest collection, if the proposer had edited his 1st post, this post wouldn't be necessary

Given $a, b,c \ge 0, a + b + c = 1$, prove that $(a^2 + b^2 + c^2)^2 + 6abc \ge ab + bc + ac$ (I. Voronovich)

$x_1=\frac{1}{2}$ and $x_{k+1}=\frac{x_k}{x_1^2+...+x_k^2}$ Prove that $\sqrt{x_k^4+4\frac{x_{k-1}}{x_{k+1}}}$ is rational

Point $D$ is marked inside a triangle $ABC$ so that $\angle ADC = \angle ABC + 60^o$, $\angle CDB =\angle CAB + 60^o$, $\angle BDA = \angle BCA + 60^o$. Prove that $AB \cdot CD = BC \cdot AD = CA \cdot BD$. (A. Levin)

Let $p$ be a positive prime integer, $S(p)$ be the number of triples $(x,y,z)$ such that $x,y,z\in\{0,1,..., p-1\}$ and $x^2+y^2+z^2$ is divided by $p$. Prove that $S(p) \ge 2p- 1$. (I. Bliznets)

Prove that there are infinitely many positive integers $n$ such that $$3^{(n-2)^{n-1}-1} -1\vdots 17n^2$$ (I. Bliznets)

Find all functions $f, g : Q \to Q$ satisfying the following equality $f(x + g(y)) = g(x) + 2 y + f(y)$ for all $x, y \in Q$. (I. Voronovich)

Let $I$ be an incenter of a triangle $ABC, A_1,B_1,C_1$ be intersection points of the circumcircle of the triangle $ABC$ and the lines $AI, BI, Cl$ respectively. Prove that a) $\frac{AI}{IA_1}+ \frac{BI}{IB_1}+ \frac{CI}{IC_1}\ge 3$ b) $AI \cdot BI \cdot CI \le I_1A_1\cdot I_2B_1 \cdot I_1C_1$ (D. Pirshtuk)

Prove that there exists a positive integer $n$ such that $n^6 + 31n^4 - 900\vdots 2009 \cdot 2010 \cdot 2011$. (I. Losev, I. Voronovich)

A positive integer $N$ is called balanced, if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. Proposed by Jorge Tipe, Peru

Find all finite sets $M \subset R, |M| \ge 2$, satisfying the following condition: for all $a, b \in M, a \ne b$, the number $a^3 - \frac{4}{9}b$ also belongs to $M$. (I. Voronovich)

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. Proposed by Hossein Karke Abadi, Iran

a) Prove that there are infinitely many pairs $(m, n)$ of positive integers satisfying the following equality $[(4 + 2\sqrt3)m] = [(4 -2\sqrt3)n]$ b) Prove that if $(m, n)$ satisfies the equality, then the number $(n + m)$ is odd. (I. Voronovich)

The following expression $x^{30} + *x^{29} +...+ *x+8 = 0$ is written on a blackboard. Two players $A$ and $B$ play the following game. $A$ starts the game. He replaces all the asterisks by the natural numbers from $1$ to $30$ (using each of them exactly once). Then player $B$ replace some of" $+$ "by ” $-$ "(by his own choice). The goal of $A$ is to get the equation having a real root greater than $10$, while the goal of $B$ is to get the equation having a real root less that or equal to $10$. If both of the players achieve their goals or nobody of them achieves his goal, then the result of the game is a draw. Otherwise, the player achieving his goal is a winner. Who of the players wins if both of them play to win? (I.Bliznets)

Numbers $a, b, c$ are the length of the medians of some triangle. If $ab + bc + ac = 1$ prove that a) $a^2b + b^2c + c^2a > \frac13$ b) $a^2b + b^2c + c^2a > \frac12$ (I. Bliznets)

Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\] Proposed by Igor Voronovich, Belarus

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. Proposed by Juhan Aru, Estonia

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. Proposed by David Monk, United Kingdom

A $50 \times 50$ square board is tiled by the tetrominoes of the following three types: Find the greatest and the smallest possible number of $L$ -shaped tetrominoes In the tiling. (Folklore)

Find the smallest value of the expression $|3 \cdot 5^m - 11 \cdot 13^n|$ for all $m,n \in N$. (Folklore)

For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied: $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$, If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$ Determine $N(n)$ for all $n\geq 2$. Proposed by Dan Schwarz, Romania

Prove that all positive real $x, y, z$ satisfy the inequality $x^y + y^z + z^x > 1$. (D. Bazylev)

The function $f : N \to N$ is defined by $f(n) = n + S(n)$, where $S(n)$ is the sum of digits in the decimal representation of positive integer $n$. a) Prove that there are infinitely many numbers $a \in N$ for which the equation $f(x) = a$ has no natural roots. b) Prove that there are infinitely many numbers $a \in N$ for which the equation $f(x) = a$ has at least two distinct natural roots. (I. Voronovich)

Prove that for positive real numbers $a, b, c$ such that $abc=1$, the following inequality holds: $$\frac{a}{b(a+b)}+\frac{b}{c(b+c)}+\frac{c}{a(c+a)} \ge \frac32$$ (I. Voronovich)

Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$. Proposed by Nikolay Beluhov, Bulgaria