MathNet 2011

Find all pairs of integers $(x, y)$ that satisfy the following equality:
$$
|x+|x+|x|| \cdot ||-y|-y|-y|=2011.
$$

Prove that if positive numbers $a$, $b$, $c$, $k$, $l$, $m$ satisfy the inequalities $abc \ge ka + lb + mc$, then they satisfy the inequality
$$
a + b + c \ge \sqrt{3}(\sqrt{k} + \sqrt{l} + \sqrt{m}).
$$

Point $M$ is the midpoint of the side $BC$ of an acute-angled triangle $ABC$, $N$ and $K$ are the feet of its altitudes $AN$ and $CK$, $H$ is the orthocenter of $ABC$. The bisector of the angle $ACB$ meets the segment $AH$ at point $T$.
Find the length of the circumradius of the triangle $NBK$ if $CT \parallel MH$, $TH = 10$.
(Slovenia, 2010)

$$
\max\{|x|-1, x^2-1\} \leq \min\{x^2-1, 1-x\},
$$
where $\max\{a,b\} = \begin{cases} a, & \text{if } a \ge b \\ b, & \text{if } a \lt b \end{cases}$, and $\min\{a,b\} = \begin{cases} b, & \text{if } a \ge b \\ a, & \text{if } a \lt b \end{cases}$.

Prove that the number $9 \cdot 2^m$ can be presented as the sum of the squares of three positive integers for any positive integer $m$.

A sequence of integers $(a_n)$ determined by
$$
a_0 = 1,\ a_1 = -1 \quad \text{and} \quad a_n = 6a_{n-1} + 5a_{n-2} \quad \text{for all } n \ge 2.
$$
Show that $a_{2012} - 2010$ is divisible by $2011$.

Given a positive integer $n$. Show that for any positive real number $x$, we have the inequality:
$$
\frac{x^n (x^{n+1} + 1)}{x^n + 1} \le \left( \frac{x+1}{2} \right)^{2n+1}
$$
When does the equality take place?

Find all four-digit perfect squares of the form $\overline{aabb}$.

Find distinct numbers $a$, $b$, $c$, $d$ such that:
$$
\frac{1}{2011} = \frac{a}{a+1} + \frac{b}{b+1} - \frac{c}{c+1} - \frac{d}{d+1}
$$

Points $A_1$ and $B_1$ are marked on the sides $AC$ and $BC$ of a triangle $ABC$ so that $AA_1 + BB_1 = AB$. Let $I$ be incenter of the triangle $ABC$, and $O$ be circumcenter of the triangle $A_1B_1C$.
Prove that the lines $OI$ and $A_1B_1$ are perpendicular.
(I. Voronovich)

Does there exist a quadruple with non parallel sides, that can be partitioned into 2011 isosceles triangles?

Given a sequence of real numbers ($x_n$):
$$
x_1 = 1 \quad \text{and} \quad x_n = \frac{2n}{(n-1)^2} \sum_{i=1}^{n-1} x_i \quad \text{for all } n \ge 2.
$$
For each positive integer $n$, let $y_n = x_{n+1} - x_n$.
Show that the sequence ($y_n$) has finite limit as $n \to +\infty$.

Let $S(n)$ be the sum of all digits in the decimal representation of $n \in \mathbb{N}$. Do there exist integers $n$ such that
$$
a)\quad n - S(n) = 3 \cdot 2010? \qquad b)\quad n - S(n) = 3 \cdot 2011?
$$
(I. Gorodnin)

$M$ and $N$ are chosen on the sides $AD$ and $BC$ of the square $ABCD$, such that $AM = BN$. Point $X$ is a feet of perpendicular from the point $D$ onto $AN$. Prove that angle $MXC$ is right.

Non-zero real numbers $a, b, c$ satisfy the equality
$$
\frac{ab}{b-c} + \frac{bc}{c-a} + \frac{ca}{a-b} = \frac{ab}{b+c} + \frac{bc}{c+a} + \frac{ca}{a+b} + 6abc.
$$
Find all possible values of the expression
$$
\frac{1}{(a^2 - b^2)^2} + \frac{1}{(b^2 - c^2)^2} + \frac{1}{(c^2 - a^2)^2}
$$

For nonnegative real numbers $a, b, c$, prove that
$$
(a+1)^2 + (b+1)^2 + (c+1)^2 \ge 3(1 + a\sqrt{b} + b\sqrt{c} + c\sqrt{a}).
$$

In triangle $ABC$ $AL$, $BM$, $CN$ are medians. Prove, that $\angle ANC = \angle ALB$ if and only if $\angle ABM = \angle LAC$.

Find all natural numbers $N$, that have two digits and are equal to the sum of their digits added to the cube of this sum.

The number $n = 11^{2011} \cdot 2011^{11}$ is given. How many divisors that are less than $n$ and are not divisors of $n$, does $n^2$ have?

Several chess players took part in a chess tournament. Each participant played exactly one game with any other participant. A participant received $1$ point for a win, $0.5$ point for a draw, and $0$ point for a lose. Any two players received different numbers of points and the participant taking the last place received $k$ points.
What is the smallest number of the wins of the participant taking the first place?