Let $A = \{1,2,..., 100\}$ and $B$ is a subset of $A$ having $48$ elements. Show that $B$ has two distint elements $x$ and $y$ whose sum is divisible by $11$.

We say that a polygon $P$ is inscribed in another polygon $Q$ when all the vertices of $P$ belong to the perimeter of $Q$. We also say in this case that $Q$ is circumscribed to $P$. Given a triangle $T$, let $\ell$ be the largest side of a square inscribed in $T$ and $L$ is the shortest side of a square circumscribed to $T$ . Find the smallest possible value of the ratio $L/\ell$ .

Real numbers $a,b$ and $c$ satisfy the equalities $2015 (a + b + c) =1$ and $ab+bc+ca=2015 abc$. Determine the numeric value of the expression $E=a^{2015}+b^{2015}+c^{2015}.$

Solve in integers $20^x+13^y=2013^z$.

Given are three pairwise externally tangent circles $ K_{1}$ , $ K_{2}$ and $ K_{3}$. denote by $ P_{1}$ tangent point of $ K_{2}$ and $ K_{3}$ and by $ P_{2}$ tangent point of $ K_{1}$ and $ K_{3}$. Let $ AB$ ($ A$ and $ B$ are different from tangency points) be a diameter of circle $ K_{3}$. Line $ AP_{2}$ intersects circle $ K_{1}$ (for second time) at point $ X$ and line $ BP_{1}$ intersects circle $ K_{2}$(for second time) at $ Y$. If $ Z$ is intersection point of lines $ AP_{1}$ and $ BP_{2}$ prove that points $ X$, $ Y$ and $ Z$ are collinear.

Players $A$ and $B$ play a game with $N \geq 2012$ coins and $2012$ boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least $1$ coin in each box. Then the two of them make moves in the order $B,A,B,A,\ldots $ by the following rules: [b](a)[/b] On every move of his $B$ passes $1$ coin from every box to an adjacent box. [b](b)[/b] On every move of hers $A$ chooses several coins that were [i]not[/i] involved in $B$'s previous move and are in different boxes. She passes every coin to an adjacent box. Player $A$'s goal is to ensure at least $1$ coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.

(1) Let $x>0,\ y$ be real numbers. For variable $t$, find the difference of Maximum and minimum value of the quadratic function $f(t)=xt^2+yt$ in $0\leq t\leq 1$. (2) Let $S$ be the domain of the points $(x,\ y)$ in the coordinate plane forming the following condition: For $x>0$ and all real numbers $t$ with $0\leq t\leq 1$ , there exists real number $z$ for which $0\leq xt^2+yt+z\leq 1$ . Sketch the outline of $S$. (3) Let $V$ be the domain of the points $(x,\ y,\ z) $ in the coordinate space forming the following condition: For $0\leq x\leq 1$ and for all real numbers $t$ with $0\leq t\leq 1$, $0\leq xt^2+yt+z\leq 1$ holds. Find the volume of $V$. [i]2011 Tokyo University entrance exam/Science, Problem 6[/i]

There are $100$ numbers on circle, and no one number is divided by other. In same time for all numbers we make next operation: If $(a,b)$ are two neighbors ($a$ is left neighbor) , then we write between $a,b$ number $\frac{a}{(a,b)}$ and erase $a,b$ This operation was repeated some times. What maximum number of $1$ we can receive ? Example: If we have circle with $3$ numbers $4,5,6$ then after operation we receive circle with numbers $\frac{4}{(4,5)}=4,\frac{5}{(5,6)}=5, \frac{6}{(6,4)}=3$.

In acute-angled triangle $ABC,$ let $M$ be the midpoint of $BC$ and let $K$ be a point on side $AB.$ We know that $AM$ meet $CK$ at $F.$ Prove that if $AK = KF$ then $AB = CF .$

Determine all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ with $ x,y \in \mathbb{R}$ such that \[ f(x \minus{} f(y)) \equal{} f(x\plus{}y) \plus{} f(y)\]

Let $ p$ be a prime number with $ p>5$. Consider the set $ X \equal{} \left\{p \minus{} n^2 \mid n\in \mathbb{N} ,\ n^2 < p\right\}$. Prove that the set $ X$ has two distinct elements $ x$ and $ y$ such that $ x\neq 1$ and $ x\mid y$. [i]Albania[/i]

Let $$f(x) = a \cos(x + 1) + b \cos(x + 2) + c \cos(x + 3)$$, where $a, b, c$ are real. Given that $f(x)$ has at least two zeros in the interval $(0, \pi)$, find all its real zeros.

A chess-board $6\times 6$ is tiled with the $2\times 1$ dominos. Prove that you can cut the board onto two parts by a straight line that does not cut dominos.

in a triangle $ABC$, $I$ is the incenter. $BI$ and $CI$ cut the circumcircle of $ABC$ at $E$ and $F$ respectively. $M$ is the midpoint of $EF$. $C$ is a circle with diameter $EF$. $IM$ cuts $C$ at two points $L$ and $K$ and the arc $BC$ of circumcircle of $ABC$ (not containing $A$) at $D$. prove that $\frac{DL}{IL}=\frac{DK}{IK}$.(25 points)

Define the function $\xi : \mathbb Z^2 \to \mathbb Z$ by $\xi(n,k) = 1$ when $n \le k$ and $\xi(n,k) = -1$ when $n > k$, and construct the polynomial \[ P(x_1, \dots, x_{1000}) = \prod_{n=1}^{1000} \left( \sum_{k=1}^{1000} \xi(n,k)x_k \right). \] (a) Determine the coefficient of $x_1x_2 \dots x_{1000}$ in $P$. (b) Show that if $x_1, x_2, \dots, x_{1000} \in \left\{ -1,1 \right\}$ then $P(x_1,x_2,\dots,x_{1000}) = 0$. [i]Proposed by Evan Chen[/i]

For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that: \[ M \leq a < b \leq c < d \leq M+49, \qquad ad=bc \] then we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number.

Matilda drew $12$ quadrilaterals. The first quadrilateral is an rectangle of integer sides and $7$ times more width than long. Every time she drew a quadrilateral she joined the midpoints of each pair of consecutive sides with a segment. It´s is known that the last quadrilateral Matilda drew was the first with area less than $1$. What is the maximum area possible for the first quadrilateral? [asy]size(200); pair A, B, C, D, M, N, P, Q; real base = 7; real altura = 1; A = (0, 0); B = (base, 0); C = (base, altura); D = (0, altura); M = (0.5*base, 0*altura); N = (0.5*base, 1*altura); P = (base, 0.5*altura); Q = (0, 0.5*altura); draw(A--B--C--D--cycle); // Rectángulo draw(M--P--N--Q--cycle); // Paralelogramo dot(M); dot(N); dot(P); dot(Q); [/asy] $\textbf{Note:}$ The above figure illustrates the first two quadrilaterals that Matilda drew.

Prove that if $n$ is large enough, in every $n\times n$ square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by $1391$.

A sequence of integers $ a_1,a_2,a_3,...$ is such that $ a_1\equal{}2, a_2\equal{}3$, and $ a_{n\plus{}1}\equal{}2a_{n\minus{}1}$ or $ 3a_n\minus{}2a_{n\minus{}1}$ for all $ n \ge 2$. Prove that no number between $ 1600$ and $ 2000$ can be an element of the sequence.

Let $n$ be a positive integer. We define $f(n)$ as the number of finite sequences $(a_1, a_2, \ldots , a_k)$ of positive integers such that $a_1 < a_2 < a_3 < \cdots < a_k$ and $$a_1+a_2^2+a_3^3+\cdots + a_k^k \leq n.$$ Determine the positive constants $\alpha$ and $C$ such that $$\lim\limits_{n\rightarrow \infty} \frac{f(n)}{n^\alpha}=C.$$

Let $n$ be a positive integer and let $V$ be a $(2n-1)$-dimensional vector space over the two-element field. Prove that for arbitrary vectors $v_1,\dots,v_{4n-1} \in V,$ there exists a sequence $1\leq i_1<\dots<i_{2n}\leq 4n-1$ of indices such that $v_{i_1}+\dots+v_{i_{2n}}=0.$

Given a parallelogram with area $1$ and we will construct lines where this lines connect a vertex with a midpoint of the side no adjacent to this vertex; with the $8$ lines formed we have a octagon inside of the parallelogram. Determine the area of this octagon

In a wrestling tournament, there are $100$ participants, all of different strengths. The stronger wrestler always wins over the weaker opponent. Each wrestler fights twice and those who win both of their fights are given awards. What is the least possible number of awardees?

Compute the number of ordered pairs $(m,n)$ of positive integers such that $(2^m-1)(2^n-1)\mid2^{10!}-1.$ [i]Proposed by Luke Robitaille[/i]

Let $ a_{0} \equal{} 1994$ and $ a_{n \plus{} 1} \equal{} \frac {a_{n}^{2}}{a_{n} \plus{} 1}$ for each nonnegative integer $ n$. Prove that $ 1994 \minus{} n$ is the greatest integer less than or equal to $ a_{n}$, $ 0 \leq n \leq 998$