SIMON-MARAIS 2018

Call a rectangle dominant if it is similar to a $2 \times 1$ rectangle. For which integers $n \geq 5$ is it possible to tile a square with $n$ dominant rectangles, which are not necessarily congruent to each other?

Ada and Byron play a game. First, Ada chooses a non-zero real number $a$ and announces it. Then Byron chooses a non-zero real number $b$ and announces it. Then Ada chooses a non-zero real number $c$ and announces it. Finally, Byron chooses a quadratic polynomial whose three coefficients are $a, b, c$ in some order.

(a) Suppose that Byron wins if the quadratic polynomial has a real root and Ada wins otherwise. Determine which player has a winning strategy.

(b) Suppose that Ada wins if the quadratic polynomial has a real root and Byron wins otherwise. Determine which player has a winning strategy.

Let $y(x)$ be the unique solution of the differential equation

$$\frac{dy}{dx} = \log_e \frac{y}{x}, \quad \text{where } x > 0 \text{ and } y > 0,$$

with the initial condition $y(1) = 2018$.

How many positive real numbers $x$ satisfy the equation $y(x) = 2000$?

For each positive integer $n$, consider a cinema with $n$ seats in a row, numbered left to right from 1 up to $n$. There is a cup holder between any two adjacent seats and there is a cup holder on the right of seat $n$. So seat 1 is next to one cup holder, while every other seat is next to two cup holders. There are $n$ people, each holding a drink, waiting in a line to sit down. In turn, each person chooses an available seat uniformly at random and carries out the following.

• If they sit next to two empty cup holders, then they place their drink in the left cup holder with probability $\frac{1}{2}$ or in the right cup holder with probability $\frac{1}{2}$.
• If they sit next to one empty cup holder, then they place their drink in that empty cup holder.
• If they sit next to zero empty cup holders, then they hold their drink in their hands.

Let $p_n$ be the probability that all $n$ people place their drink in a cup holder.

Determine $p_1 + p_2 + p_3 + \cdots$.

For all positive integers $n$ and real numbers $x_1, x_2, \ldots, x_n$, prove that

$$\sum_{i=1}^{n} \sum_{j=1}^{n} \min(i, j) x_i x_j \geq 0.$$

(We define $\min(a, b) = a$ if $a \leq b$ and $\min(a, b) = b$ if $a > b$.)

Let $S$ be the set of real numbers that can be expressed as $\sqrt{m} - \sqrt{n}$, where $m$ and $n$ are positive integers.

Do there exist real numbers $a < b$ such that the open interval $(a, b)$ contains only finitely many elements of $S$?

Three spiders try to catch a beetle in a game. They are all initially positioned on the edges of a regular dodecahedron whose edges have length 1. At some point in time, they start moving continuously along the edges of the dodecahedron. The beetle and one of the spiders move with maximum speed 1, while the remaining two spiders move with maximum speed $\frac{1}{2018}$. Each player always knows their own position and the position of every other player. A player can turn around at any moment and can react to the behaviour of other players instantaneously. The spiders can communicate to decide on a strategy before and during the game. If any spider occupies the same position as the beetle at some time, then the spiders win the game.

Prove that the spiders can win the game, regardless of the initial positions of all players and regardless of how the beetle moves.

(A regular dodecahedron is a convex polyhedron with twelve faces, each of which is a pentagon with equal side lengths and equal angles. Three faces meet at each vertex.)

The following problem is open in the sense that no solution is currently known. An explicit expression with proof for the known case of $|A(5, n)|$ will be awarded 2 points. Further progress on the problem may be awarded more points.

For positive integers $m$ and $n$, let $A(m, n)$ be the set of $2 \times mn$ matrices $M$ with entries from the set $\{1, 2, \ldots, m\}$ such that

• each of the numbers $1, 2, \ldots, m$ appears exactly $2n$ times;
• $M_{1,1} \leq M_{1,2} \leq \cdots \leq M_{1,mn}$ and $M_{2,1} \leq M_{2,2} \leq \cdots \leq M_{2,mn}$; and
• $M_{1,j} < M_{2,j}$ for $j = 1, 2, \ldots, mn$.

Determine $|A(m, n)|$.