Let $a_1 < a_2 < a_3 < \ldots$ be an infinite strictly increasing sequence of positive integers such that for some positive integer $k$ and all integers $n>k$ the term $a_n$ is a sum of two members of the sequence. Prove that the sequence contains infinitely many composite numbers.

Let $ABC$ ($AC > BC$) be an acute triangle with midpoint $M$ of $AB$ and altitude $CD$ ($D \in AB$). Let $AE \perp CM$ ($E\in CM$) and $F$ be the midpoint of $CD$. Prove that $FM$ is tangent to the circumcircle of $EMB$.

The function $f:\mathbb{R} \to \mathbb{R}$ is such that $f(x+y) \geq f(x) + yf(f(x))$ for all $x,y \in \mathbb{R}$. Prove that: a) $f(f(x)) \leq 0$ for all $x \in \mathbb{R}$; b) if $f(0) \geq 0$, then $f(x) = 0$ for all $x\in \mathbb{R}$

Prove that for every positive integer $m$ there are infinitely many pairs $(x,y)$ of coprime positive integers such that $x$ divides $y^2 + m$ and $y$ divides $x^2 + m$.

Let $ABCD$ be a cyclic quadrilateral. The lines $AD$ and $BC$ intersect at $P$ and the lines $AB$ and $CD$ intersect at $Q$. If $\angle APQ = 90^{\circ}$, prove that the perpendicular from $P$ to $AB$ bisects the diagonal $BD$.

Prove that for any positive integer $n\geq 3$ there exist $n$ distinct positive integers, the sum of cubes of which is also a perfect cube.

In a country there are $15$ cities, some pairs of which are connected by a single two-way airline of a company. There are $3$ companies and if any of them cancels all its flights, then it would still be possible to reach every city from every other city using the other two companies. At least how many two-way airlines are there?