Amy and Bob play the game. At the beginning, Amy writes down a positive integer on the board. Then the players take moves in turn, Bob moves first. On any move of his, Bob replaces the number $n$ on the blackboard with a number of the form $n-a^2$, where $a$ is a positive integer. On any move of hers, Amy replaces the number $n$ on the blackboard with a number of the form $n^k$, where $k$ is a positive integer. Bob wins if the number on the board becomes zero. Can Amy prevent Bob’s win? [i]Maxim Didin, Russia[/i]

Let $A$ be a set of $16$ positive integers with the property that the product of any two distinct members of $A$ will not exceed 1994. Show that there are numbers $a$ and $b$ in the set $A$ such that the gcd of $a$ and $b$ is greater than 1.

Let $p_1,p_2, p_3$ be positive numbers such that $p_1 + p_2 + p_3 = 1$. If $a_1 <a_2 <a_3$ and $b_1 <b_2 <b_3$ prove that $$(a_1p_1 + a_2p_2 + a_3p_3) (b_1p_1 + b_2p_2 + b_3p_3)\le (a_1b_1p_1 + a_2b_2p_2 + a_3b_3p_3)$$

In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.

A line $l$ is drawn through the intersection point $H$ of altitudes of acute-angle triangles. Prove that symmetric images $l_a, l_b, l_c$ of $l$ with respect to the sides $BC,CA,AB$ have one point in common, which lies on the circumcircle of $ABC.$

Given two lines in a plane, intersecting at an acute angle. In the direction of one of the straight lines, compression is performed with a coefficient of 1/2. Prove that there is a point from which the distance to the point of intersection of the lines increases. Note: What is meant here is a transformation in which each point moves parallel to one straight line so that its distance to the second straight line is halved, while it remains the same side from the second straight line. [hide=original wording] На плоскости даны две прямые, пересекающиеся под острым углом. В направлении одной из прямых производится сжатие 1 с коэффициентом 1/2. Доказать, что найдется точка, расстояние от которой до точки пересечения прямых увеличится. Здесь имеется в виду преобразование, при котором каждая точка перемещается параллельно одной прямой так, что её расстояние до второй прямой уменьшается вдвое, причём она остаётся по ту же самую сторону от второй прямой[/hide]

A line $ l$ is tangent to a circle $ S$ at $ A$. For any points $ B,C$ on $ l$ on opposite sides of $ A$, let the other tangents from $ B$ and $ C$ to $ S$ intersect at a point $ P$. If $ B,C$ vary on $ l$ so that the product $ AB \cdot AC$ is constant, find the locus of $ P$.

Let $ABC $ be an acute triangle such that $AB\neq AC$ ,with circumcircle $ \Gamma$ and circumcenter $O$. Let $M$ be the midpoint of $BC$ and $D$ be a point on $ \Gamma$ such that $AD \perp BC$. let $T$ be a point such that $BDCT$ is a parallelogram and $Q$ a point on the same side of $BC$ as $A$ such that $\angle{BQM}=\angle{BCA}$ and $\angle{CQM}=\angle{CBA}$. Let the line $AO$ intersect $ \Gamma$ at $E$ $(E\neq A)$ and let the circumcircle of $\triangle ETQ$ intersect $ \Gamma$ at point $X\neq E$. Prove that the point $A,M$ and $X$ are collinear.

Find a $n\in\mathbb{N}$ such that for all primes $p$, $n$ is divisible by $p$ if and only if $n$ is divisible by $p-1$.

For every positive integer $n$, denote by $D_n$ the number of permutations $(x_1, \dots, x_n)$ of $(1,2,\dots, n)$ such that $x_j\neq j$ for every $1\le j\le n$. For $1\le k\le \frac{n}{2}$, denote by $\Delta (n,k)$ the number of permutations $(x_1,\dots, x_n)$ of $(1,2,\dots, n)$ such that $x_i=k+i$ for every $1\le i\le k$ and $x_j\neq j$ for every $1\le j\le n$. Prove that $$\Delta (n,k)=\sum_{i=0}^{k=1} \binom{k-1}{i} \frac{D_{(n+1)-(k+i)}}{n-(k+i)}$$ (Proposed by Combinatorics; Ferdowsi University of Mashhad, Iran; Mirzavaziri)

Rectangle on a checked paper with length of a unit square side being $1$ Is divided into domino figures( two unit square sharing a common edge). Prove that you colour all corners of squares on the edge of rectangle and inside rectangle with $3$ colours such that for any two corners with distance $1$ the following conditions hold: they are coloured in different colour if the line connecting the two corners is on the border of two domino figures and coloured in same colour if the line connecting the two corners is inside a domino figure.

Consider positive real numbers $x,y,z$. Prove the inequality $$\frac 1x+\frac 1y+\frac 1z+\frac{9}{x+y+z}\ge 3\left (\left (\frac{1}{2x+y}+\frac{1}{x+2y}\right )+\left (\frac{1}{2y+z}+\frac{1}{y+2z}\right )+\left (\frac{1}{2z+x}+\frac{1}{x+2z}\right )\right ).$$ [i] (Vlad Robu \& Sergiu Novac)[/i]

Show that there exists a positive integer $ k$ such that $ k \cdot 2^{n} \plus{} 1$ is composite for all $ n \in \mathbb{N}_{0}$.

In the figure, a semicircle is folded along the $ AN $ string and intersects the $ MN $ diameter in $ B $. $ MB: BN = 2: 3 $ and $ MN = 10 $ are known to be. If $ AN = x $, what is the value of $ x ^ 2 $?

Determine the number of functions $f: \mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N} \to \mathbb{N}$ such that for all $n \in \mathbb{N}$: \[ f(g(n)) = n + 2015, \] \[ g(f(n)) = n^2 + 2015. \]

Given four points $ A_1, A_2, A_3, A_4$ in the plane, no three collinear, such that \[ A_1A_2 \cdot A_3 A_4 \equal{} A_1 A_3 \cdot A_2 A_4 \equal{} A_1 A_4 \cdot A_2 A_3, \] denote by $ O_i$ the circumcenter of $ \triangle A_j A_k A_l$ with $ \{i,j,k,l\} \equal{} \{1,2,3,4\}.$ Assuming $ \forall i A_i \neq O_i ,$ prove that the four lines $ A_iO_i$ are concurrent or parallel. [i]Nikolai Ivanov Beluhov, Bulgaria[/i]