Let $f(x)=x^{n}+5x^{n-1}+3$, where $n>1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

Let $n$ be a positive integer. We know that the set $I_n = \{ 1, 2,\ldots , n\}$ has exactly $2^n$ subsets, so there are $8^n$ ordered triples $(A, B, C)$, where $A, B$, and $C$ are subsets of $I_n$. For each of these triples we consider the number $\mid A \cap B \cap C\mid$. Prove that the sum of the $8^n$ numbers considered is a multiple of $n$. Clarification: $\mid Y\mid$ denotes the number of elements in the set $Y$.

Which one divides $2^{2^{2010}}+2^{2^{2009}}+1$? $ \textbf{(A)}\ 19 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ \text{None} $

Solve the following equation in the set of integer numbers: \[ x^{2010}-2006=4y^{2009}+4y^{2008}+2007y. \]

(a) Two players take turns taking $1, 2$ or $3$ stones at random from a given set of $3$ piles, in which initially on $11, 22$ and $33$ stones. If after the move of one of the players in any two groups the same number of stones will remain, this player has won. Who will win with the right game of both players? (b) Two players take turns taking $1$ or $2$ stones from one pile, randomly selected from a given set of $3$ ordered piles, in which at first $100, 200$ and $300$ stones, in order from left to right. Additionally it is forbidden to make a course at which, for some pair of the next handfuls, quantity of stones in the left will be more than the number of stones in the right. If after the move of one of the players of the stones in handfuls will not remain, then this player won. Who will win with the right game of both players? [hide=original wording] 1. Два гравця по черзi беруть 1, 2 чи 3 камiнця довiльним чином з заданого набору з 3 купок, в яких спочатку по 11, 22 i 33 камiнцiв. Якщо пiсля хода одного з гравцiв в якихось двух купках залишиться однакова кiлькiсть камiнцiв, то цей гравець виграв. Хто виграє при правильнiй грi обох гравцiв? 2. Два гравця по черзi беруть 1 чи 2 камiнця з одної купки, довiльної вибраної з заданого набору з 3 впорядкованих купок, в яких спочатку по 100, 200 i 300 камiнцiв, в порядку злiва направо. Додатково забороняется робити ход при якому, для деякої пари сусiднiх купок, кiлькiсть камiнцiв в лiвiй стане бiльше нiж кiлькiсть камiнцiв в правiй. Якщо пiсля ходу одного з гравцiв камiнцiв в купках не залишиться, то цей гравець виграв. Хто виграє при правильнiй грi обох гравцiв?[/hide]

Let $a, b$ be positive integers such that $54^a=a^b$. Prove that $a$ is a power of $54$.

A rectangle with integer side lengths has the property that its area minus $5$ times its perimeter equals $2023$. Find the minimum possible perimeter of this rectangle.

(A. Myakishev, 8--9) In the plane, given two concentric circles with the center $ A$. Let $ B$ be an arbitrary point on some of these circles, and $ C$ on the other one. For every triangle $ ABC$, consider two equal circles mutually tangent at the point $ K$, such that one of these circles is tangent to the line $ AB$ at point $ B$ and the other one is tangent to the line $ AC$ at point $ C$. Determine the locus of points $ K$.

Say that a function $f : \{1, 2, . . . , 1001\} \to Z$ is [i]almost [/i] polynomial if there is a polynomial $p(x) = a_{200}x^{200} +... + a_1x + a_0$ such that each an is an integer with $|a_n| \le 201$, and such that $|f(x) - p(x)| \le 1$ for all $x \in \{1, 2, . . . , 1001\}$. Let $N$ be the number of almost polynomial functions. Compute the remainder upon dividing $N$ by $199$.

Evaluate \[\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{(1+\cos x)\{1-\tan ^ 2 \frac{x}{2}\tan (x+\sin x)\tan (x-\sin x)\}}{\tan (x+\sin x)}\ dx\]

Let $a,b,c$ be positive reals satisfying $a^3+b^3+c^3+abc=4$. Prove that \[ \frac{(5a^2+bc)^2}{(a+b)(a+c)} + \frac{(5b^2+ca)^2}{(b+c)(b+a)} + \frac{(5c^2+ab)^2}{(c+a)(c+b)} \ge \frac{(a^3+b^3+c^3+6)^2}{a+b+c} \] and determine the cases of equality. [i]Proposed by Evan Chen[/i]

Let $x_1=1/20$, $x_2=1/13$, and \[x_{n+2}=\dfrac{2x_nx_{n+1}(x_n+x_{n+1})}{x_n^2+x_{n+1}^2}\] for all integers $n\geq 1$. Evaluate $\textstyle\sum_{n=1}^\infty(1/(x_n+x_{n+1}))$.

Determine all (not necessarily finite) sets $S$ of points in the plane such that given any four distinct points in $S$, there is a circle passing through all four or a line passing through some three. [i]Carl Lian.[/i]

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

How many rectangles are there in the diagram below such that the sum of the numbers within the rectangle is a multiple of 7? [asy] int n; n=0; for (int i=0; i<=7;++i) { draw((i,0)--(i,7)); draw((0,i)--(7,i)); for (int a=0; a<=7;++a) { if ((a != 7)&&(i != 7)) { n=n+1; label((string) n,(a,i),(1.5,2)); } } } [/asy]

Suppose that the differentiable functions $a, b, f, g:\mathbb{R} \rightarrow \mathbb{R} $ satisfy \[ f(x)\geq 0, f'(x) \geq 0,g(x)\geq 0, g'(x) \geq 0 \text{ for all } x \in \mathbb{R}, \] \[\lim_{x\rightarrow \infty} a(x)=A\geq 0,\lim_{x\rightarrow \infty} b(x)=B\geq 0, \lim_{x\rightarrow \infty} f(x)=\lim_{x\rightarrow \infty} g(x)=\infty,\] and \[\frac{f'(x)}{g'(x)}+a(x)\frac{f(x)}{g(x)}=b(x).\] Prove that $\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=\frac{B}{A+1}$.