SIMON-MARAIS 2017

The five sides and five diagonals of a regular pentagon are drawn on a piece of paper. Two people play a game, in which they take turns to colour one of these ten line segments. The first player colours line segments blue, while the second player colours line segments red. A player cannot colour a line segment that has already been coloured. A player wins if they are the first to create a triangle in their own colour, whose three vertices are also vertices of the regular pentagon. The game is declared a draw if all ten line segments have been coloured without a player winning.

Determine whether the first player, the second player, or neither player can force a win.

Let $a_1, a_2, a_3, \ldots$ be the sequence of real numbers defined by $a_1 = 1$ and

$$a_m = \frac{1}{a_1^2 + a_2^2 + \cdots + a_{m-1}^2}$$

for $m \geq 2$.

Determine whether there exists a positive integer $N$ such that

$$a_1 + a_2 + \cdots + a_N > 2017^{2017}$$.

For each positive integer $n$, let $M(n)$ be the $n \times n$ matrix whose $(i,j)$ entry is equal to 1 if $i + 1$ is divisible by $j$, and equal to 0 otherwise.

Prove that $M(n)$ is invertible if and only if $n + 1$ is square-free.

(An integer is square-free if it is not divisible by the square of an integer larger than 1.)

Let $A_1, A_2, \ldots, A_{2017}$ be the vertices of a regular polygon with 2017 sides.

Prove that there exists a point $P$ in the plane of the polygon such that the vector

$$\sum_{k=1}^{2017} k \frac{\overrightarrow{PA_k}}{\|\overrightarrow{PA_k}\|^5}$$

is the zero vector.

(The notation $\|\overrightarrow{XY}\|$ represents the length of the vector $\overrightarrow{XY}$.)

Maryam labels each vertex of a tetrahedron with the sum of the lengths of the three edges meeting at that vertex. She then observes that the labels at the four vertices of the tetrahedron are all equal.

For each vertex of the tetrahedron, prove that the lengths of the three edges meeting at that vertex are the three side lengths of a triangle.

Determine all pairs $(p, q)$ of positive integers such that $p$ and $q$ are prime, and $p^{q-1} + q^{p-1}$ is the square of an integer.

Each point in the plane with integer coordinates is coloured red or blue such that the following two properties hold.

• For any two red points, the line segment joining them does not contain any blue points.
• For any two blue points that are distance 2 apart, the midpoint of the line segment joining them is blue.

Prove that if three red points are the vertices of a triangle, then the interior of the triangle does not contain any blue points.

The following problem is open in the sense that no solution is currently known. Progress on the problem may be awarded points. An example of progress on the problem is a non-trivial bound on the sequence defined below.

For each integer $n \geq 2$, consider a regular polygon with $2n$ sides, all of length 1. Let $C(n)$ denote the number of ways to tile this polygon using quadrilaterals whose sides all have length 1.

Determine the limit inferior and the limit superior of the sequence defined by

$$\frac{1}{n^2} \log_2 C(n).$$