SIMON-MARAIS 2023

For $0 < q < 1$, the $q$-Fibonacci spiral is constructed as described below. An arc of radius 1 is first drawn inside a $1 \times 1$ square. A second arc is then drawn in a $q \times q$ square, then a third in a $q^2 \times q^2$ square, and so on ad infinitum, to create a continuous curve. The example on the left shows the case $q = \frac{1}{2}$, while the example on the right shows $q = \frac{\sqrt{5}-1}{2}$.

Prove that there exists a circle centred at the centre of the initial $1 \times 1$ square such that for each $0 < q < 1$, the limiting endpoint of the $q$-Fibonacci spiral lies on this circle.

Let $n$ be a positive integer and let $f_1(x), \ldots, f_n(x)$ be affine functions from $\mathbb{R}$ to $\mathbb{R}$ such that, amongst the $n$ graphs of these functions, no two are parallel and no three are concurrent. Let $S$ be the set of all convex functions $g(x)$ from $\mathbb{R}$ to $\mathbb{R}$ such that for each $x \in \mathbb{R}$, there exists $i$ such that $g(x) = f_i(x)$.

Determine the largest and smallest possible values of $|S|$ in terms of $n$.

(A function $f(x)$ is affine if it is of the form $f(x) = ax + b$ for some $a, b \in \mathbb{R}$. A function $g(x)$ is convex if $g(\lambda x + (1 - \lambda)y) \leq \lambda g(x) + (1 - \lambda)g(y)$ for all $x, y \in \mathbb{R}$ and $0 \leq \lambda \leq 1$.)

For each positive integer $n$, let $f(n)$ denote the smallest possible value of

$$|A_1 \cup A_2 \cup \cdots \cup A_n|,$$

where $A_1, A_2, \ldots, A_n$ are sets such that $A_i \not\subseteq A_j$ and $|A_i| \neq |A_j|$ whenever $i \neq j$.

Determine $f(n)$ for each positive integer $n$.

Let $x_0, x_1, x_2, \ldots$ be a sequence of positive real numbers such that for all $n \geq 0$,

$$x_{n+1} = \frac{(n^2 + 1)x_n^2}{x_n^3 + n^2}.$$

For which values of $x_0$ is this sequence bounded?

Find the smallest positive real number $r$ with the following property: For every choice of 2023 unit vectors $v_1, v_2, \ldots, v_{2023} \in \mathbb{R}^2$, a point $p$ can be found in the plane such that for each subset $S$ of $\{1, 2, \ldots, 2023\}$, the sum

$$\sum_{i \in S} v_i$$

lies inside the disc $\{x \in \mathbb{R}^2 : ||x - p|| \leq r\}$.

There are 256 players in a tennis tournament who are ranked from 1 to 256, with 1 corresponding to the highest rank and 256 corresponding to the lowest rank. When two players play a match in the tournament, the player whose rank is higher wins the match with probability $\frac{3}{5}$.

In each round of the tournament, the player with the highest rank plays against the player with the second highest rank, the player with the third highest rank plays against the player with the fourth highest rank, and so on. At the end of the round, the players who win proceed to the next round and the players who lose exit the tournament. After eight rounds, there is one player remaining in the tournament and they are declared the winner.

Determine the expected value of the rank of the winner.

Let $n$ be a positive integer. Let $A, B$ and $C$ be three $n$-dimensional subspaces of $\mathbb{R}^{2n}$ with the property that $A \cap B = B \cap C = C \cap A = \{0\}$. Prove that there exist bases $\{a_1, a_2, \ldots, a_n\}$ of $A$, $\{b_1, b_2, \ldots, b_n\}$ of $B$ and $\{c_1, c_2, \ldots, c_n\}$ of $C$ with the property that for each $i \in \{1, 2, \ldots, n\}$, the vectors $a_i, b_i$ and $c_i$ are linearly dependent.

The following problem is open in the sense that the answer to part (b) is not currently known. A solution to part (a) will be awarded 7 points. Up to 7 additional points may be awarded for progress on part (b).

(a) Let $n$ be a positive integer that is not a perfect square. Find all pairs $(a, b)$ of positive integers for which there exists a positive real number $r$, such that

$$r^a + \sqrt{n} \quad \text{and} \quad r^b + \sqrt{n}$$

are both rational numbers.

(b) Let $n$ be a positive integer that is not a perfect square. Find all pairs $(a, b)$ of positive integers for which there exists a real number $r$, such that

$$r^a + \sqrt{n} \quad \text{and} \quad r^b + \sqrt{n}$$

are both rational numbers.