Problem 1. Given a cube $ABCD.EFGH$ with an edge with length 4 units and $P$ be the midpoint/center of side $EFGH$. If $M$ is the midpoint of $PH$, determine the length of segment $AM$. Problem 2. Find all reals $k$ such that the system of equations \begin{align*} a^2 + ab &= kb^2 \\ b^2 + bc &= kc^2 \\ c^2 + ca &= ka^2 \end{align*}have (a) real positive solution(s) $a, b, c$. Problem 3. Each cell of a checkerboard with size $m \times n$ is colored with either black or white, such that: (a) The number of black and white cells in each row are the same. (b) If a row intersects a column at some black cell, then said row and column contain the same number of black tiles. (c) If a row intersects a column at some white cell, then said row and column contain the same number of white tiles. Determine all possible values of $m$ and $n$ so that the coloring above can be done. Problem 4. Determine all nonnegative integers $k$ such that we can always find a noninteger positive real $x$ which satisfies \[ \lfloor x + k \rfloor^{\lfloor x + k \rfloor} = \lceil x \rceil^{\lfloor x \rfloor} + \lfloor x \rfloor^{\lceil x \rceil}. \] Problem 5. On a triangle $ABC$ where $AC > BC$, with $O$ as its circumcenter. Let $M$ be the circumcircle of $ABC$ such that $CM$ is the bisector of $\angle{ACB}$/ Let $\Gamma$ be the circle with diameter $CM$. The bisectors of $BOC$ and $AOC$ intersect $\Gamma$ respectively at points $P$ and $Q$. If $K$ is the midpoint of $CM$, prove that points $P, Q, O, K$ are concyclic.