IMO 2016 Day 1

I'm surprised nobody else has posted this. Yes, there are plenty of discussions in Art of Problem Solving, but why not here? And of course there are more problems.

Problem 1

Triangle $BCF$ has a right angle at $B$. Let $A$ be the point on line $CF$ such that $FA = FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen so that $DA = DC$ and $AC$ is the bisector of $\angle{DAB}$. Point $E$ is chosen so that $EA = ED$ and $AD$ is the bisector of $\angle{EAC}$. Let $M$ be the midpoint of $CF$. Let $X$ be the point such that $AMXE$ is a parallelogram. Prove that $BD$, $FX$, and $ME$ are concurrent.

Problem 2

Find all positive integers $n$ for which each cell of an $n \times n$ table can be filled with one of the letters I, M, and O in such a way that:

• in each row and each column, one third of the entries are I, one third are M, and one third are O; and
• in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are I, one third are M, and one third are O.

Note. The rows and columns of an $n \times n$ table are each labelled $1$ to $n$ in a natural order. Thus each cell corresponds to a pair of positive integer $(i, j)$ with $1 \le i,j \le n$. For $n > 1$, the table has $4n-2$ diagonals of two types. A diagonal of first type consists all cells $(i, j)$ for which $i+j$ is a constant, and the diagonal of this second type consists all cells $(i, j)$ for which $i-j$ is constant.

Problem 3

Let $P = A_1 A_2 \cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$. Prove that $2S$ is an integer divisible by $n$.