Let $\mathbb{Z}^+$ denote the set of positive integers. Find all functions $f: \mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m,m)=m$ $f(m,k) = f(k,m)$ $f(m, m+k) = f(m,k)$ , for each $m,k \in \mathbb{Z}^+$.

A positive integer is called a "double number" if its decimal representation consists of a block of digits, not commencing with $0$, followed immediately by an identical block. So, for instance, $360360$ is a double number, but $36036$ is not. Show that there are infinitely many double numbers which are perfect squares.

Let $C_1$ and $C_2$ be given circles. Let $A_1$ on $C_1$ and $A_2$ on $C_2$ be fixed points. If chord $A_1P_1$ of $C_1$ is parallel to chord $A_2P_2$ of $C_2$, find the locus of the midpoint of $P_1P_2$.

There is a stone on each square of $n\times n$ chessboard. We gather $n^2$ stones and distribute them to the squares (again each square contains one stone) such that any two adjacent stones are again adjacent. Find all distributions such that at least one stone at the corners remains at its initial square. (Two squares are adjacent if they share a common edge.)

There are $n\geq2$ weights such that each weighs a positive integer less than $n$ and their total weights is less than $2n$. Prove that there is a subset of these weights such that their total weights is equal to $n$.

The circle, which is tangent to the circumcircle of isosceles triangle $ABC$ ($AB=AC$), is tangent $AB$ and $AC$ at $P$ and $Q$, respectively. Prove that the midpoint $I$ of the segment $PQ$ is the center of the excircle (which is tangent to $BC$) of the triangle .