IMC 2022

Let $f \colon [0, 1] \to (0, \infty)$ be an integrable function such that $f(x) \cdot f(1 - x) = 1$ for all $x \in [0, 1]$. Prove that

Let $n$ be a positive integer. Find all $n \times n$ real matrices $A$ with only real eigenvalues satisfying

for some integer $k \ge n$.

($A^T$ denotes the transpose of $A$.)

Let $p$ be a prime number. A flea is staying at point $0$ of the real line. At each minute, the flea has three possibilities: to stay at its position, or to move by $1$ to the left or to the right. After $p - 1$ minutes, it wants to be at $0$ again. Denote by $f(p)$ the number of its strategies to do this (for example, $f(3) = 3$: it may either stay at $0$ for the entire time, or go to the left and then to the right, or go to the right and then to the left). Find $f(p)$ modulo $p$.

Let $n > 3$ be an integer. Let $\Omega$ be the set of all triples of distinct elements of $\{1, 2, \ldots, n\}$. Let $m$ denote the minimal number of colours which suffice to colour $\Omega$ so that whenever $1 \le a < b < c < d \le n$, the triples $\{a, b, c\}$ and $\{b, c, d\}$ have different colours. Prove that

We colour all the sides and diagonals of a regular polygon $P$ with $43$ vertices either red or blue in such a way that every vertex is an endpoint of $20$ red segments and $22$ blue segments. A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same colour. Suppose that there are $2022$ blue monochromatic triangles. How many red monochromatic triangles are there?

Let $p > 2$ be a prime number. Prove that there is a permutation $(x_1, x_2, \ldots, x_{p-1})$ of the numbers $(1, 2, \ldots, p-1)$ such that

Let $A_1, A_2, \ldots, A_k$ be $n \times n$ idempotent complex matrices such that

Prove that at least one of the given matrices has rank $\le \frac{n}{k}$.

(A matrix $A$ is called idempotent if $A^2 = A$.)

Let $n, k \ge 3$ be integers, and let $S$ be a circle. Let $n$ blue points and $k$ red points be chosen uniformly and independently at random on the circle $S$. Denote by $F$ the intersection of the convex hull of the red points and the convex hull of the blue points. Let $m$ be the number of vertices of the convex polygon $F$ (in particular, $m = 0$ when $F$ is empty). Find the expected value of $m$.