p1. Determine which of the following numbers is greater : $99!$ or $50^{99}$. p2. Let $x , y$ be positive real numbers with $x > y$ that satisfy the relation $x^2 +y^2 = axy$ where $a$ is a real number greater than $2$. Find all possible values of $a$ that make $\frac{x + y}{x - y}$ an integer. p3. In the figure below $ABCD$ is a square. Segment $AE$ is $3$ units and segment $EF$ is $1$ unit. The angles $\angle AED$ and $\angle BFA$ are right. Calculate the length of the segments $FC$ p4. A design $X$ is an array of the digits $1,2,..., 9$ in the shape of an $X$, for example, We will say that a design $X$ is balanced if the sum of the numbers of each of the diagonals match. Determine the number of designs $X$ that are balanced.