Let $ABCD$ be a convex quadrilateral and $M$ be the intersection of its diagonals. Through $M$ draw a line meeting the side $AB$ at $P$ and the side $CD$ at $Q$. Find all the quadrilaterals so that there exists the segment $PQ$ that divides the triangles $ABM$ and $CDM$ into 4 similar triangles.
Let $I$ be the incenter of $\triangle ABC$ and $\omega$ be its incircle. Let $D$ be the intersection of $\omega$ with $BC$. Suppose $X,Y$ are points on $\omega$ such that lines $IX,IY$ and $AD$ are tangent to the circumcircle of $\triangle AXY$. Prove that line $XY$ bisects $BC$.
Find all functions $f:\mathbb Z^+\to\mathbb Z^+$, where $\mathbb Z^+$ is the set of all positive integers, such that for any positive integers $m,n$,
$$f^{f(m)}(n)+n=f^m(n)+f(n).$$
Let $a_1,a_2,\ldots,a_n$ be positive numbers and $A$ their arithmetic mean. Prove that
$$A^{n-1}\ge\frac1n\sum_{i=1}^n a_1\cdots\widehat{a_i}\cdots a_n,$$
where $\widehat{a_i}$ denotes that the term $a_i$ is omitted.
Find the smallest positive $m$ so that it is possible to choose 8 vertices from a regular polygon with $m$ sides such that no three of them are the vertices of an isosceles triangle.