Let $n$ be a positive integer. There are $n$ colors available. Each of the integers from $1$ to $1000$ must be painted with one of the $n$ colors such that any two different numbers, if one divides the other, are painted in different colors. Determine the smallest value of $n$ for which this is possible.

A red equilateral triangle $T$ with side length $1$ is drawn on a plane. For a positive real $c$, we place three blue equilateral triangle shaped paper with side length $c$ on a plane to cover $T$ completely. Find the minimum value of $c$. As shown in the picture, it doesn't matter if the blue papers overlap each other or stick out from $T$. Folding or tearing the paper is not allowed.

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

Some of the vertices of a convex $n$-gon are connected by segments, such that any two of them have no common interior point. Prove that, for any $n$ points in general position, there exists a one-to-one correspondence between the points and the vertices of the $n$ gon, such that any two segments between the points, corresponding to the respective segments from the $n$ gon, have no common interior point.

Three points $X, Y,Z$ are on a striaght line such that $XY = 10$ and $XZ = 3$. What is the product of all possible values of $YZ$?

Let be a prime number $ p, $ the quotient ring $ R=\mathbb{Z}[X,Y]/(pX,pY), $ and a prime ideal $ I\supset pA $ that is not maximal. Show that the ring $ \left\{ r/i|r\in R, i\in I \right\} $ is factorial.

Alex dissects a paper triangle into two triangles. Each minute after this he dissects one of obtained triangles into two triangles. After some time (at least one hour) it appeared that all obtained triangles were congruent. Find all initial triangles for which this is possible.

For any positive integer $n$, let $s(n)$ be the sum of its digits, written in decimal base. Prove that for each integer $n \ge 1$ there exists a positive integer $x$ such that the fraction $\frac{x + k}{s(x + k)}$ is not integral, for each integer $k$ with $0 \le k \le n$.

Let $ABC$ be a triangle and its incircle meets $BC, AC, AB$ at $D, E$ and $F$ respectively. Let point $ P $ on the incircle and inside $ \triangle AEF $. Let $ X=PB \cap DF , Y=PC \cap DE, Q=EX \cap FY $. Prove that the points $ A$ and $Q$ lies on $DP$ simultaneously or located opposite sides from $DP$.

$7$ pupils has been given $20$ candies, $5$ candies of $4$ different kinds, so that each pupil has no more then one candy of each kind. Prove that there are two pupils that have three or more pairs of candies of the same kind.

A polynomial $f (x)$ with real coefficients is called [i]completely reducible[/i] if it is a product of at least two non-constant polynomials whose coefficientsare all nonnegative real numbers. Show: If $f (x^{2011})$ is completely reducible, then $f(x)$ is also.

Determine all positive integers $n$ such that $$n\cdot 2^{n-1}+1$$ is a perfect square.

How many ordered pairs of sets $(A, B)$ have the properties: 1. $ A\subseteq \{1, 2, 3, 4, 5, 6\} $ 2. $ B\subseteq\{2, 3, 4, 5, 6, 7, 8\} $ 3. $ A\cap B $ has exactly $3$ elements.

Several jugs (not necessarily of the same size) with juices are placed along a circle. It is allowed to transfuse any part of juice (maybe nothing or the total content) from any jug to the neighboring one on the right, so that the latter one is not overflowed and the sugariness of its content becomes equal to $10\%$. It is known that at the initial moment such transfusion is possible from each jug. Prove that it is possible to perform several transfusions in some order, at most one transfusion from each jug, such that the sugariness of the content of each non-empty jug will become equal to $10\%$. (Sugariness is the percent of sugar in a jug, by weight. Sugar is always uniformly distributed in a jug.)

The positive real numbers $x,y,z$ satisfy $xy+yz+zx=1$. Prove that: $$ 2(x^2+y^2+z^2)+\frac{4}{3}\bigg (\frac{1}{x^2+1}+\frac{1}{y^2+1}+\frac{1}{z^2+1}\bigg) \ge 5 $$

Find all isosceles triangles that cannot be cut into three isosceles triangles with the same sides.

In a triangle $ABC$, the median $AD$ divides $\angle{BAC}$ in the ratio $1:2$. Extend $AD$ to $E$ such that $EB$ is perpendicular $AB$. Given that $BE=3,BA=4$, find the integer nearest to $BC^2$.

Michael and Andrew are playing the game Bust, which is played as follows: Michael chooses a positive integer less than or equal to $99$, and writes it on the board. Andrew then makes a move, which consists of him choosing a positive integer less than or equal to $ 8$ and increasing the integer on the board by the integer he chose. Play then alternates in this manner, with each person making exactly one move, until the integer on the board becomes greater than or equal to $100$. The person who made the last move loses. Let S be the sum of all numbers for which Michael could choose initially and win with both people playing optimally. Find S.

Prove that a necessary and sufficient for the existence of a set $ S \subset \{1,2,...,n \}$ with the property that the integers $ 0,1,...,n\minus{}1$ all have an odd number of representations in the form $ x\minus{}y, x,y \in S$, is that $ (2n\minus{}1)$ has a multiple of the form $ 2.4^k\minus{}1$ [i]L. Lovasz, J. Pelikan[/i]

Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$f(f(x)-9f(y))=(x+3y)^2f(x-3y)$$ for all $x,y\in \mathbb{R}$.

Find the set of solutions to the equation $\log_{\lfloor x\rfloor}(x^2-1)=2$

Let $S$ be a finite set of polynomials in two variables, $x$ and $y$. For $n$ a positive integer, define $ \Omega _ { n } ( S ) $ to be the collection of all expressions $ p _ { 1 } p _ { 2 } \dots p _ { k } ,$ where $p_i \in S$ and $1\leq k \leq n$. Let $d_n(S)$ indicate the maximum number of linearly independent polynomials in $ \Omega _ { n } ( S ) $. For example, $ \Omega _ { 2 } \left( \left\{ x ^ { 2 } , y \right\} \right) = \left\{ x ^ { 2 } , y , x ^ { 2 } y , x ^ { 4 } , y ^ { 2 } \right\} $ and $d _ { 2 } \left( \left\{ x ^ { 2 } , y \right\} \right) = 5 $ (a) Find $ d _ { 2 } ( \{ 1 , x , x + 1 , y \} ) $. (b) Find a closed formula in $n$ for $ d _ { n } ( \{ 1 , x , y \} ) $. (c) Calculate the least upper bound over all such sets of $ \overline{\text{lim}} _ { n \rightarrow \infty } \frac { \log d _ { n } ( S ) } { \log n } $ ($ \overline{\text{lim}} _ { n \rightarrow \infty } a _ { n } = \lim _ { n \rightarrow \infty } ( \sup \left\{ a _ { n } , a _ { n + 1 } , \ldots \right\} $, where sup means supremum or least upper bound.)

Let $a$ and $b$ be two non-zero rational numbers such that the equation $ax^2+by^2=0$ has a non-zero solution in rational numbers . Prove that for any rational number $t$ , there is a solution of the equation $ax^2+by^2=t$.

Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations: \begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*}

Show that for $n \geq 5$, the integers $1, 2, \ldots n$ can be split into two groups so that the sum of the integers in one group equals the product of the integers in the other group.