Two players take turns playing on a $3\times1001$ board whose squares are initially all white. Each player, in his turn, paints two squares located in the same row or column black, not necessarily adjacent. The player who cannot make his move loses the game. Determine which of the two players has a strategy that allows them to win, no matter how well his opponent plays.

There are $101$ positive integers $a_1, a_2, \ldots, a_{101}$ such that for every index $i$, with $1 \leq i \leq 101$, $a_i+1$ is a multiple of $a_{i+1}$. Find the greatest possible value of the largest of the $101$ numbers.

Let $ABC$ be an acute triangle. The point $B'$ of the line $CA$ is such that $A$, $C$, $B'$ are in that order on the line and $B'C=AB$; the point $C'$ of the line $AB$ is such that $A$, $B$, $C'$ are in that order on the line and $C'B=AC$. Show that the circumcenter of triangle $AB'C'$ belongs to the circumcircle of triangle $ABC$.

Find the least possible value of $\dfrac{(x^2+1)(4y^2+1)(9z^2+1)}{6xyz}$ if $x$, $y$, $z$ are not necessarily distinct positive real numbers.

In chess, a knight placed on a chess board can move by jumping to an adjacent square in one direction (up, down, left, or right) then jumping to the next two squares in a perpendicular direction. We then say that a square in a chess board can be attacked by a knight if the knight can end up on that square after a move. Thus, depending on where a knight is placed, it can attack as many as eight squares, or maybe even less. In a $10 \times 10$ chess board, what is the maximum number of knights that can be placed such that each square on the board can be attacked by at most one knight?

Find all pairs of positive integers $(n, k)$ that satisfy the equation $$n!+n=n^k$$