IMC 2024

Determine all pairs $(a, b) \in \mathbb{C} \times \mathbb{C}$ satisfying

For $n = 1, 2, \ldots$ let

where $\log$ denotes the natural logarithm. Find $\lim\limits_{n \to \infty} S_n$.

For which positive integers $n$ does there exist an $n \times n$ matrix $A$ whose entries are all in $\{0, 1\}$, such that $A^2$ is the matrix of all ones?

Let $g$ and $h$ be two distinct elements of a group $G$, and let $n$ be a positive integer. Consider a sequence $w = (w_1, w_2, \ldots)$ which is not eventually periodic and where each $w_i$ is either $g$ or $h$. Denote by $H$ the subgroup of $G$ generated by all elements of the form $w_k w_{k+1} \ldots w_{k+n-1}$ with $k \ge 1$. Prove that $H$ does not depend on the choice of the sequence $w$ (but may depend on $n$).

Let $n > d$ be positive integers. Choose $n$ independent, uniformly distributed random points $x_1, \ldots, x_n$ in the unit ball $B \subset \mathbb{R}^d$ centered at the origin. For a point $p \in B$ denote by $f(p)$ the probability that the convex hull of $x_1, \ldots, x_n$ contains $p$. Prove that if $p, q \in B$ and the distance of $p$ from the origin is smaller than the distance of $q$ from the origin, then $f(p) \ge f(q)$.

Prove that for any function $f \colon \mathbb{Q} \to \mathbb{Z}$, there exist $a, b, c \in \mathbb{Q}$ such that $a < b < c$, $f(b) \ge f(a)$, and $f(b) \ge f(c)$.

Let $n$ be a positive integer. Suppose that $A$ and $B$ are invertible $n \times n$ matrices with complex entries such that $A + B = I$ (where $I$ is the identity matrix) and

Find all possible values of $\det(AB)$ for the given $n$.

Define the sequence $x_1, x_2, \ldots$ by the initial terms $x_1 = 2$, $x_2 = 4$, and the recurrence relation

Prove that $\lim\limits_{n \to \infty} \dfrac{x_n}{2^n}$ exists and satisfies

A matrix $A = (a_{ij})$ is called nice, if it has the following properties:

(i) the set of all entries of $A$ is $\{1, 2, \ldots, 2t\}$ for some integer $t$;

(ii) the entries are non-decreasing in every row and in every column: $a_{i,j} \le a_{i,j+1}$ and $a_{i,j} \le a_{i+1,j}$;

(iii) equal entries can appear only in the same row or the same column: if $a_{i,j} = a_{k,\ell}$, then either $i = k$ or $j = \ell$;

(iv) for each $s = 1, 2, \ldots, 2t - 1$, there exist $i \ne k$ and $j \ne \ell$ such that $a_{i,j} = s$ and $a_{k,\ell} = s + 1$.

Prove that for any positive integers $m$ and $n$, the number of nice $m \times n$ matrices is even.

For example, the only two nice $2 \times 3$ matrices are $\begin{pmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \end{pmatrix}$ and $\begin{pmatrix} 1 & 1 & 3 \\ 2 & 4 & 4 \end{pmatrix}$.

We say that a square-free positive integer $n$ is almost prime if

for all integers $x$, where $1 = d_1 < d_2 < \ldots < d_k = n$ are all the positive divisors of $n$. Suppose that $r$ is a Fermat prime (i.e. it is a prime of the form $2^{2^m} + 1$ for an integer $m \ge 0$), $p$ is a prime divisor of an almost prime integer $n$, and $p \equiv 1 \pmod{r}$. Show that, with the above notation, $d_i \equiv 1 \pmod{r}$ for all $1 \le i \le k$.

(An integer $n$ is called square-free if it is not divisible by $d^2$ for any integer $d > 1$.)