Determine the minimum value of $\dfrac{m^m}{1\cdot 3\cdot 5\cdot \ldots \cdot(2m-1)}$ for positive integers $m$.

Find all positive integers $a\in \{1,2,3,4\}$ such that if $b=2a$, then there exist infinitely many positive integers $n$ such that $$\underbrace{aa\dots aa}_\textrm{$2n$}-\underbrace{bb\dots bb}_\textrm{$n$}$$ is a perfect square.

The cells of an $n \times n$ table are filled with the numbers $1,2,\dots,n$ for the first row, $n+1,n+2,\dots,2n$ for the second, and so on until $n^2-n,n^2-n+1,\dots,n^2$ for the $n$-th row. Peter picks $n$ numbers from this table such that no two of them lie on the same row or column. Peter then calculates the sum $S$ of the numbers he has chosen. Prove that Peter always gets the same number for $S$, no matter how he chooses his $n$ numbers.

Let $n$ be a positive integer. A \emph{pseudo-Gangnam Style} is a dance competition between players $A$ and $B$. At time $0$, both players face to the north. For every $k\ge 1$, at time $2k-1$, player $A$ can either choose to stay stationary, or turn $90^{\circ}$ clockwise, and player $B$ is forced to follow him; at time $2k$, player $B$ can either choose to stay stationary, or turn $90^{\circ}$ clockwise, and player $A$ is forced to follow him. After time $n$, the music stops and the competition is over. If the final position of both players is north or east, $A$ wins. If the final position of both players is south or west, $B$ wins. Determine who has a winning strategy when: (a) $n=2013^{2012}$ (b) $n=2013^{2013}$

Consider a triangle $ABC$ with height $AH$ and $H$ on $BC$. Let $\gamma_1$ and $\gamma_2$ be the circles with diameter $BH,CH$ respectively, and let their centers be $O_1$ and $O_2$. Points $X,Y$ lie on $\gamma_1,\gamma_2$ respectively such that $AX,AY$ are tangent to each circle and $X,Y,H$ are all distinct. $P$ is a point such that $PO_1$ is perpendicular to $BX$ and $PO_2$ is perpendicular to $CY$. Prove that the circumcircles of $PXY$ and $AO_1O_2$ are tangent to each other.