$101$ distinct numbers are chosen among the integers between $0$ and $1000$. Prove that, among the absolute values ​​of their pairwise differences, there are ten different numbers not exceeding $100$.

Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians speak at most three languages, prove that there are at least three of the mathematicians who can speak the same language.

Let $ A$ be a subset of $ \{ 0,1,2,...,1997 \}$ containing more than $ 1000$ elements. Prove that either $ A$ contains a power of $ 2$ (that is, a number of the form $ 2^k$ with $ k\equal{}0,1,2,...)$ or there exist two distinct elements $ a,b \in A$ such that $ a\plus{}b$ is a power of $ 2$.

Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

Let $n\geq 1$ be an integer and let $X$ be a set of $n^2+1$ positive integers such that in any subset of $X$ with $n+1$ elements there exist two elements $x\neq y$ such that $x\mid y$. Prove that there exists a subset $\{x_1,x_2,\ldots, x_{n+1} \} \in X$ such that $x_i \mid x_{i+1}$ for all $i=1,2,\ldots, n$.

Prove that among $81$ natural numbers whose prime divisors are in the set $\{2, 3, 5\}$ there exist four numbers whose product is the fourth power of an integer.

In an archery competition of 30 contestants, the target is divided into two zones, zone 1 and zone 2. Each arrow hitting the zone 1 gets 10 points, when hitting zone 2 will get 5 points and no score for miss. Each contestant throws 16 arrows. At the end of the competition, the statistics show that more than 50% of the arrows hit zone 2. The number of arrows that hit zone 1 is equal to the arrows which are missed. Prove than there are two contestants having equal score.

Let $a,b$ be positive real numbers. For any positive integer $n$, denote by $x_n$ the sum of digits of the number $[an+b]$ in it's decimal representation. Show that the sequence $(x_n)_{n\ge 1}$ contains a constant subsequence. [i]Laurentiu Panaitopol[/i]

For $m$ a positive integer, let $s(m)$ be the sum of the digits of $m$. For $n\ge 2$, let $f(n)$ be the minimal $k$ for which there exists a set $S$ of $n$ positive integers such that $s\left(\sum_{x\in X} x\right)=k$ for any nonempty subset $X\subset S$. Prove that there are constants $0<C_1<C_2$ with \[C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.\]

Given $101$ distinct non-negative integers less than $5050$ show that one can choose four $a, b, c, d$ such that $a + b - c - d$ is a multiple of $5050$

There are four basketball players $A,B,C,D$. Initially the ball is with $A$. The ball is always passed from one person to a different person. In how many ways can the ball come back to $A$ after $\textbf{seven}$ moves? (for example $A\rightarrow C\rightarrow B\rightarrow D\rightarrow A\rightarrow B\rightarrow C\rightarrow A$, or $A\rightarrow D\rightarrow A\rightarrow D\rightarrow C\rightarrow A\rightarrow B\rightarrow A)$.

Let $k>1$ be an integer, set $n=2^{k+1}$. Prove that for any positive integers $a_1<a_2<\cdots<a_n$, the number $\prod_{1\leq i<j\leq n}(a_i+a_j)$ has at least $k+1$ different prime divisors.

A group consist of n tourists. Among every 3 of them there are 2 which are not familiar. For every partition of the tourists in 2 buses you can find 2 tourists that are in the same bus and are familiar with each other. Prove that is a tourist familiar to at most $\displaystyle \frac 2{5}n$ tourists.

Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20. [i]Proposed by Luxembourg[/i]

The terms of the sequence $ (a_i)$ defined by $ a_{n \plus{} 2} \equal{} \frac {a_n \plus{} 2009} {1 \plus{} a_{n \plus{} 1}}$ for $ n \ge 1$ are positive integers. Find the minimum possible value of $ a_1 \plus{} a_2$.

Let $S$ be a set of $n$ points in the plane such that any two points of $S$ are at least $1$ unit apart. Prove there is a subset $T$ of $S$ with at least $\frac{n}{7}$ points such that any two points of $T$ are at least $\sqrt{3}$ units apart.

Let $n\ge 1$ be an odd integer. Alice and Bob play the following game, taking alternating turns, with Alice playing first. The playing area consists of $n$ spaces, arranged in a line. Initially all spaces are empty. At each turn, a player either • places a stone in an empty space, or • removes a stone from a nonempty space $s,$ places a stone in the nearest empty space to the left of $s$ (if such a space exists), and places a stone in the nearest empty space to the right of $s$ (if such a space exists). Furthermore, a move is permitted only if the resulting position has not occurred previously in the game. A player loses if he or she is unable to move. Assuming that both players play optimally throughout the game, what moves may Alice make on her first turn?

a) Prove that there exist integers $a, b, c$ not all zero and each of absolute value less than one million, such that $$ |a +b \sqrt{2} +c \sqrt{3} | <10^{-11} .$$ b) Let $ a, b, c$ be integers, not all zero and each of absolute value less than one million. Prove that $$ |a +b \sqrt{2} +c \sqrt{3} | >10^{-21} .$$

Let $k$ be an odd number that is greater than or equal to $3$. Prove that there exists a $k^{th}$-degree integer-valued polynomial with non-integer-coefficients that has the following properties: (1) $f(0)=0$ and $f(1)=1$; and. (2) There exist infinitely many positive integers $n$ so that if the following equation: \[ n= f(x_1)+\cdots+f(x_s), \] has integer solutions $x_1, x_2, \dots, x_s$, then $s \geq 2^k-1$.

How many ways to fill the board $ 4\times 4$ by nonnegative integers, such that sum of the numbers of each row and each column is 3?

There are 51 senators in a senate. The senate needs to be divided into $n$ committees so that each senator is on one committee. Each senator hates exactly three other senators. (If senator A hates senator B, then senator B does [i]not[/i] necessarily hate senator A.) Find the smallest $n$ such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee.

Let $S$ be a set of $10$ distinct positive real numbers. Show that there exist $x,y \in S$ such that \[0 < x - y < \frac{(1 + x)(1 + y)}{9}.\]

For any positive integer $m$,show that there exist integers $a,b$ satisfying $\left | a \right |\leq m$, $ \left | b \right |\leq m$, $0< a+b\sqrt{2}\leq \frac{1+\sqrt{2}}{m+2}$

Prove that there exist integers $a$, $b$, $c$ with $1 \le a < b < c \le 25$ and \[ S(a^6+2014) = S(b^6+2014) = S(c^6+2014) \] where $S(n)$ denotes the sum of the decimal digits of $n$. [i]Proposed by Evan Chen[/i]

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]