Line intersects hyperbola $H_1$, given by the equation $y=1/x$ at points $A$ and $B$, and hyperbola $H_2$, given by the equation $y=-1/x$ at points $C$ and $D$. Tangents to hyperbola $H_1$ at points $A$ and $B$ intersect at point $M$, and tangents to hyperbola $H_2$ at points $C$ and $D$ intersect at point $N$. Prove that points $M$ and $N$ are symmetric about the origin.

A natural number $n$ was alternately divided by $29$, $41$ and $59$. The result was three nonzero remainders, the sum of which equals $n$. Find all such $n$

Let $A_1$ be a midmoint of $BC$, and $G$ is a centroid of the non-isosceles triangle $\triangle ABC$. $GBKL$ and $GCMN$ are the squares lying on the left with respect to rays $GB$ and $GC$ respectively. Let $A_2$ be a midpoint of a segment connecting the centers of the squares $GBKL$ and $GCMN$. Circumcircle of triangle $\triangle A_{1}A_{2}G$ intersects $BC$ at points $A_1$ and $X$. Find $\frac{A_{1}X}{XH}$, where $H$ is a base of altitude $AH$ of the triangle $\triangle ABC$.

Find all functions $f(x)$ determined on interval $[0,1]$, satisfying following conditions $$\{f(x)\}\sin^{2}x+\{x\}\cos f(x)\cos x=f(x)$$ $$f(f(x))=f(x)$$ Here $\{y\}$ means a fractional part of number $y$

Find all real $x\geq-1$ such that for all $a_1,...,a_n\geq1$, where $n\geq2$ the following inequality holds $$\frac{a_1+x}{2}*\frac{a_2+x}{2}*...*\frac{a_n+x}{2}\leq\frac{a_1a_2...a_n+x}{2}$$

Let $M$ be a set of natural numbers from $1$ to $2015$ which are not perfect squares. a) Prove that for any $n\in M$ $\{\sqrt{n}\}\geq 0.011$ b) Prove that there exists number $n\in M$ such that $\{\sqrt{n}\}<0.0115$ Here $\{y\}$ means the fractional part of number $y$

Let $I$ be an incenter of a triangle $\triangle ABC$. Points $A_1, B_1, C_1$ are the tangent points of the inscribed circle on sides $BC$, $CA$ and $AB$ respectively. Circumcircle of $\triangle BC_1B_1$ intersects line $BC$ at points $B$ and $K$ and Circumcircle of $\triangle CB_1C_1$ intersects line $BC$ at points $C$ and $L$. Prove that lines $LC_1$, $KB_1$ and $IA_1$ are concurrent.

Let $n$ be a natural number. What is the least number $m$ ($m>n$) such that the set of all natural numbers forn $n$ to $m$ (inclusively) can be divided into subsets such that in each subset one of the numbers equals the sum of other numbers in this subset?