Suppose that a sequence $x_1,x_2,\ldots,x_{2001}$ of positive real numbers satisfies $$3x^2_{n+1}=7x_nx_{n+1}-3x_{n+1}-2x^2_n+x_n\enspace\text{ and }\enspace x_{37}=x_{2001}.$$Find the maximum possible value of $x_1$.

In an acute-angled triangle $ABC$, $a,b,c$ are sides, $m_a,m_b,m_c$ the corresponding medians, $R$ the circumradius and $r$ the inradius. Prove the inequality $$\frac{a^2+b^2}{a+b}\cdot\frac{b^2+c^2}{b+c}\cdot\frac{a^2+c^2}{a+c}\ge16R^2r\frac{m_a}a\cdot\frac{m_b}b\cdot\frac{m_c}c.$$

Let $a,b$ be coprime positive integers with $a$ even and $a>b$. Show that there exist infinitely many pairs $(m,n)$ of coprime positive integers such that $m\mid a^{n-1}-b^{n-1}$ and $n\mid a^{m-1}-b^{m-1}$.

On a line are given $n>3$ points. Find the number of colorings of these points in red and blue, such that in any set of consequent points the difference between the numbers of red and blue points does not exceed $2$.

Let $A,B,C,D,E,F$ be the midpoints of consecutive sides of a hexagon with parallel opposite sides. Prove that the points $AB\cap DE$, $BC\cap EF$, $AC\cap DF$ lie on a line.

Some cells of a $10\times10$ board are marked so that each cell has an even number of neighboring (i.e. sharing a side) marked cells. Find the maximum possible number of marked cells.

Prove that for every positive integer $n$ there exists a polynomial $p(x)$ of degree $n$ with real coefficients, having $n$ distinct real roots and satisfying $$p(x)p(4-x)=p(x(4-x))$$

For positive real numbers $b_1,b_2,\ldots,b_n$ define $$a_1=\frac{b_1}{b_1+b_2+\ldots+b_n}\enspace\text{ and }\enspace a_k=\frac{b_1+\ldots+b_k}{b_1+\ldots+b_{k-1}}\text{ for }k>1.$$Prove that $a_1+a_2+\ldots+a_n\le\frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_n}$

Let $k\ge0$ be a given integer. Suppose there exists positive integer $n,d$ and an odd integer $m>1$ with $d\mid m^{2^k}-1$ and $m\mid n^d+1$. Find all possible values of $\frac{m^{2^k}-1}d$.

Some cells of a $2n\times2n$ board are marked so that each cell has an even number of neighboring (i.e. sharing a side) marked cells. Find the number of such markings.

Chords $AC$ and $BD$ of a circle $w$ intersect at $E$. A circle that is internally tangent to $w$ at a point $F$ also touches the segments $DE$ and $EC$. Prove that the bisector of $\angle AFB$ passes through the incenter of $\triangle AEB$.

In a tennis tournament, any two of the $n$ participants played a match (the winner of a match gets $1$ point, the loser gets $0$). The scores after the tournament were $r_1\le r_2\le\ldots\le r_n$. A match between two players is called wrong if after it the winner has a score less than or equal to that of the loser. Consider the set $I=\{i|r_1\ge i\}$. Prove that the number of wrong matches is not less than $\sum_{i\in I}(r_i-i+1)$, and show that this value is realizable