Let $ABC$ be an acute-angled triangle with $AB\neq AC$. The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ respectively. Denote by $O$ the midpoint of the side $BC$. The bisectors of the angles $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the side $BC$.
Find all polynomials $P$ with real coefficients such that for all reals $a$, $b$, $c$ such that $ab+bc+ca = 0$, we have $$
P(a-b) + P(b-c) + P(c-a) = 2P(a+b+c).
$$
Define a “hook” to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure.
Determine all $m\times n$ rectangles that can be covered by hooks without gaps or overlaps, such that no part of a hook covers area outside the rectangle.
Let $n \ge 3$ be an integer and $t_1$, $t_2$, …, $t_n$ positive real numbers such that $$
n^2+1 \gt \left(t_1 + t_2 + \dots + t_n\right)
\left( \frac{1}{t_1} + \frac{1}{t_2} + \dots + \frac{1}{t_n} \right).
$$ Show that $t_i$, $t_j$, $t_k$ are the sides of a triangle for all $i$, $j$, $k$ with $1 \le i \lt j \lt k \le n$.
In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies $$
\angle PBC=\angle DBA \quad\text{and}\quad \angle PDC=\angle BDA.
$$ Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$.
We call a positive integer *alternating* if every two consecutive digits in its decimal representation are of different parity. Find all positive integers $n$ which have an alternating multiple.
