SIMON-MARAIS 2024

Let $a, b, c$ be real numbers greater than 1 satisfying

$$\lfloor a \rfloor b = \lfloor b \rfloor c = \lfloor c \rfloor a.$$

Prove that $a = b = c$.

(Here, $\lfloor x \rfloor$ denotes the largest integer that is less than or equal to $x$.)

A positive integer $n$ is tripairable if it is possible to partition the set $\{1, 2, \ldots, n\}$ into disjoint pairs such that the sum of the two elements in each pair is a power of 3. For example, 6 is tripairable because $\{1, 2, 3, 4, 5, 6\} = \{1, 2\} \cup \{3, 6\} \cup \{4, 5\}$ and

$$1 + 2 = 3^1, \quad 3 + 6 = 3^2 \quad \text{and} \quad 4 + 5 = 3^2$$

are all powers of 3.

How many positive integers less than or equal to 2024 are tripairable?

Let $W$ be a fixed positive integer. Let $S$ be the set of all pairs $(a, b)$ of positive integers such that $a \neq b$. For each $(a, b) \in S$, let $m(a, b)$ be the largest integer satisfying $m(a, b) \leq \frac{1 + na}{1 + nb}$ for all integers $n \geq 1$.

(a) For each $(a, b) \in S$, prove that there exists a positive integer $k$ such that

$$m(a, b) \leq \frac{1 + na}{W + nb},$$

for all $n \geq k$.

(b) For each $(a, b) \in S$, let $k(a, b)$ be the smallest value of $k$ that satisfies the condition of part (a). Determine $\max\{k(a, b) \mid (a, b) \in S\}$ or prove that it does not exist.

Define a sequence by $s_0 = 1$ and for $d \geq 1$, $s_d = s_{d-1} + X_d$, where $X_d$ is chosen uniformly at random from the set $\{1, 2, \ldots, d\}$.

What is the probability that the sequence $s_0, s_1, s_2, \ldots$ contains infinitely many primes?

Alice has six boxes labelled 1 through 6. She secretly chooses exactly two of the boxes and places a coin inside each. Bob is trying to guess which two boxes contain the coins. Each time Bob guesses, he does so by tapping exactly two of the boxes. Alice then responds by telling him the total number of coins inside the two boxes that he tapped. Bob successfully finds the two coins when Alice responds with the number 2.

What is the smallest positive integer $n$ such that Bob can always find the two coins in at most $n$ guesses?

Determine all continuous functions $f : \mathbb{R} \setminus \{1\} \rightarrow \mathbb{R}$ that satisfy

$$f(x) = (x + 1)f(x^2),$$

for all $x \in \mathbb{R} \setminus \{-1, 1\}$.

Let $\mathcal{L}$ be the set of all lines in the plane and let $\mathcal{P}$ be the set of all points in the plane. Determine whether there exists a function $g : \mathcal{L} \rightarrow \mathcal{P}$ such that for any two distinct non-parallel lines $\ell_1, \ell_2 \in \mathcal{L}$, we have

(a) $g(\ell_1) \neq g(\ell_2)$, and

(b) if $\ell_3$ is the line passing through $g(\ell_1)$ and $g(\ell_2)$, then $g(\ell_3)$ is the intersection of $\ell_1$ and $\ell_2$.

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 $n$ be an odd positive integer and let

$$f_n(x, y, z) = x^n + y^n + z^n + (x + y + z)^n.$$

(a) Prove that there exist infinitely many values of $n$ such that

$$f_n(x, y, z) \equiv (x + y)(y + z)(z + x)g(x, y, z)h(x, y, z) \pmod{2},$$

for some integer polynomials $g(x, y, z)$ and $h(x, y, z)$, neither of which is constant modulo 2.

(b) Determine all values of $n$ such that

$$f_n(x, y, z) \equiv (x + y)(y + z)(z + x)g(x, y, z)h(x, y, z) \pmod{2},$$

for some integer polynomials $g(x, y, z)$ and $h(x, y, z)$, neither of which is constant modulo 2.

(Two integer polynomials are congruent modulo 2 if every coefficient of their difference is even. A polynomial is constant modulo 2 if it is congruent to a constant polynomial modulo 2.)