p1. In how many ways can $2019$ be written as the sum of consecutive positive integers? p2. On a table there are $2019$ chips. Two players take turns drawing chips. The first player to play can draw any odd number of chips between $ 1$ and $99$. The other player can draw any even number of chips between$ 2$ and $100$. The player who can not continue playing loses. Determine if any of the players has a winning strategy. p3. Consider a rectangle $ABCD$ with $| AB | > | BC |$ and let $E$ be the midpoint of $CD$ side. $F$ is chosen in $CD$ such that $| CF | = | BC |$. Suppose $AC \perp BE$. Prove that $| AB | = | BF |$. p4. Each face of a cube of dimensions $1000\times 1000\times 1000$ is divided into $1000^2$ unit squares of $1\times 1$. Determine the largest number of little squares units that can be marked on the cube in such a way that no pair of them share a side in common.