An acute triangle $ABC$ in which $AB<AC$ is given. The bisector of $\angle BAC$ crosses $BC$ at $D$. Point $M$ is the midpoint of $BC$. Prove that the line though centers of circles escribed on triangles $ABC$ and $ADM$ is parallel to $AD$.

A subset $S$ of size $n$ of a plane consisting of points with both coordinates integer is given, where $n$ is an odd number. The injective function $f\colon S\rightarrow S$ satisfies the following: for each pair of points $A, B\in S$, the distance between points $f(A)$ and $f(B)$ is not smaller than the distance between points $A$ and $B$. Prove there exists a point $X$ such that $f(X)=X$.

Find all real numbers $c$ for which there exists a function $f\colon\mathbb R\rightarrow \mathbb R$ such that for each $x, y\in\mathbb R$ it's true that $$f(f(x)+f(y))+cxy=f(x+y).$$

Let $n$ be a positive integer. Suppose there are exactly $M$ squarefree integers $k$ such that $\left\lfloor\frac nk\right\rfloor$ is odd in the set $\{ 1, 2,\ldots, n\}$. Prove $M$ is odd. An integer is squarefree if it is not divisible by any square other than $1$.

An acute triangle $ABC$ in which $AB<AC$ is given. Points $E$ and $F$ are feet of its heights from $B$ and $C$, respectively. The line tangent in point $A$ to the circle escribed on $ABC$ crosses $BC$ at $P$. The line parallel to $BC$ that goes through point $A$ crosses $EF$ at $Q$. Prove $PQ$ is perpendicular to the median from $A$ of triangle $ABC$.

A prime $p>3$ is given. Let $K$ be the number of such permutations $(a_1, a_2, \ldots, a_p)$ of $\{ 1, 2, \ldots, p\}$ such that $$a_1a_2+a_2a_3+\ldots + a_{p-1}a_p+a_pa_1$$is divisible by $p$. Prove $K+p$ is divisible by $p^2$.