Let $H$ be the orthocenter of the triangle $ABC$. Let $M$ and $N$ be the midpoints of the sides $AB$ and $AC$, respectively. Assume that $H$ lies inside the quadrilateral $BMNC$ and that the circumcircles of triangles $BMH$ and $CNH$ are tangent to each other. The line through $H$ parallel to $BC$ intersects the circumcircles of the triangles $BMH$ and $CNH$ in the points $K$ and $L$, respectively. Let $F$ be the intersection point of $MK$ and $NL$ and let $J$ be the incenter of triangle $MHN$. Prove that $FJ=FA$.
Let
$$f(x)=\frac1x+\frac1{x-2}+\frac1{x-4}+\cdots+\frac1{x-2018},\qquad g(x)=\frac1{x-1}+\frac1{x-3}+\frac1{x-5}+\cdots+\frac1{x-2017}.$$
Prove that $|f(x)-g(x)|\gt 2$ for every non-integer real number $x$ satisfying $0\lt x\lt 2018$.
A collection of $n$ squares on the plane is called tri-connected if:
(i) all squares are congruent;
(ii) if two squares have a point $P$ in common, then $P$ is a vertex of each square;
(iii) each square touches exactly three other squares.
How many positive integers $n$ with $2018\le n\le3018$ admit such a collection?
Let $ABC$ be an equilateral triangle. From $A$ draw a ray toward the interior. Whenever it reaches a side, it reflects according to the law of reflection: arriving at directed angle $\alpha$, it leaves at directed angle $180^\circ-\alpha$. After $n$ bounces the ray returns to $A$ without ever landing on either of the other two vertices. Find all possible $n$.
Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s,t$, if $P(s)$ and $P(t)$ are integers, then $P(st)$ is also an integer.
