SIMON-MARAIS 2022

Let $ABCD$ be a unit square, and let $P$ be a point inside triangle $ABC$. Let $E$ and $F$ be the points on $AB$ and $AD$, respectively, such that $AEPF$ is a rectangle. Let $x$ and $y$ denote the lengths of $AE$ and $AF$, respectively.

Determine a two-variable polynomial $f(x,y)$ with the property that the area of $AEPF$ is greater than the area of each of the quadrilaterals $EBCP$ and $FPCD$ if and only if $f(x,y) > 0$.

Let $n$ be a positive integer, and let $S$ be a set with $2^n$ elements. Let $A_1, A_2, \ldots, A_n$ be randomly and independently chosen subsets of $S$, where each possible subset of $S$ is chosen with equal probability. Let $P_n$ be the probability that

$$A_1 \cup A_2 \cup \cdots \cup A_n = S \quad \text{and} \quad A_1 \cap A_2 \cap \cdots \cap A_n = \emptyset.$$

Prove that $\lim_{n \to \infty} P_n = \frac{1}{e^2}$.

Let $0 < a < 1$ be a fixed real number. Show that there are at least two values of $x$ in the interval $(0,1)$ such that

$$\int_0^x \left( \sin \left( \frac{\pi \sin \frac{\pi t}{2}}{2} \right) + \frac{2}{\pi} \arcsin \left( \frac{2}{\pi} \arcsin t \right) - 2t \right) \mathrm{d}t = \frac{1}{2} \left( a - \frac{2}{\pi} \arcsin \left( \frac{2}{\pi} \arcsin a \right) \right) \left( \sin \left( \frac{\pi \sin \frac{\pi a}{2}}{2} \right) - a \right).$$

Let $n$ be a positive integer, and let $q \geq 3$ be an odd integer such that every prime factor of $q$ is larger than $n$. Prove that

$$\frac{1}{n! (q-1)^n} \prod_{i=1}^{n} (q^i - 1)$$

is an integer that has no prime factor in common with $\frac{q-1}{2}$.

Let $A > 1$ be a real number. Determine all pairs $(m, n)$ of positive integers for which there exists a positive real number $x$ such that $(1 + x)^m = (1 + Ax)^n$.

Let $a, b,$ and $c$ be real numbers, and let $P$ be the polynomial

$$P(x) = x^6 + ax^4 + bx^2 + c.$$

Suppose that there is a unique circle $\Gamma$ in the complex plane such that all of the roots of $P$ lie on $\Gamma$. Prove that $b^3 = a^3c$.

Ari and Sam are playing a game in which they take turns breaking a block of chocolate in two and eating one of the pieces. At each stage of the game the block of chocolate is a rectangle with integer side lengths. On each player’s turn, they break the block of chocolate into two such rectangles along a horizontal or vertical line, and eat the piece with smaller area. (If the two pieces have the same area they may eat either one.) The game ends when the block of chocolate is a $1 \times 1$ rectangle, and the winner is the last player to take their turn breaking the chocolate in two.

At the start of the game the block of chocolate is a $58 \times 2022$ rectangle. If Ari goes first, which player has a winning strategy?

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).

Let $a_0, a_1, a_2, \ldots$ and $b_0, b_1, b_2, \ldots$ be two sequences of positive integers, satisfying $a_0b_0 \geq 600$ and

$$a_{n+1} = a_n + 2 \cdot \lfloor b_n/20 \rfloor,$$

$$b_{n+1} = b_n + 3 \cdot \lfloor a_n/30 \rfloor,$$

for all $n \geq 0$.

(a) Prove that there exists a nonnegative integer $N$ such that

$$-13 \leq a_n - b_n \leq 23,$$

for all $n \geq N$.

(b) Must there exist a nonnegative integer $n$ such that $a_n = b_n$?

Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.