$M$ is an integer set with a finite number of elements. Among any three elements of this set, it is always possible to choose two such that the sum of these two numbers is an element of $M.$ How many elements can $M$ have at most?

Let $(a_n)_n\geq 0$ and $a_{m+n}+a_{m-n}=\frac{1}{2}(a_{2m}+a_{2n})$ for every $m\geq n\geq0.$ If $a_1=1,$ then find the value of $a_{2007}.$

In $\bigtriangleup$$ABC$ $BL$ is bisector. Arbitrary point $M$ on segment $CL$ is chosen. Tangent to $\odot$$(ABC)$ at $B$ intersects $CA$ at $P$. Tangents to $\odot$$BLM$ at $B$ and $M$ intersect at point $Q$. Prove that $PQ$$\parallel$$BL$.

Find all positive integers $m$ and $n$ such that $(2^{2^{n}}+1)(2^{2^{m}}+1) $ is divisible by $m\cdot n $ .

Given an equilateral triangle $ABC$ and a point $P$ so that the distances $P$ to $A$ and to $C$ are not farther than the distances $P$ to $B$. Prove that $PB = PA + PC$ if and only if $P$ lies on the circumcircle of $\vartriangle ABC$.

Four numbers are written on the board: $1, 3, 6, 10.$ Each time two arbitrary numbers, $a$ and $b$ are deleted, and numbers $a + b$ and $ab$ are written in their place. Is it possible to get numbers $2015, 2016, 2017, 2018$ on the board after several such operations?

The degree of the polynomial $P(x)$ is $2017.$ Prove that the number of distinct real roots of the equation $P(P(x)) = 0$ is not less than the number of distinct real roots of the equation $P(x) = 0.$

Find all natural numbers a, b such that $ a^{n}+ b^{n} = c^{n+1}$ where c and n are naturals.