IMC 2022

Let $f: [0,1] \to (0, \infty)$ be an integrable function such that $f(x)f(1-x) = 1$ for all $x\in [0,1]$ . Prove that $\int_0^1f(x)dx \geq 1$ .

For a positive integer $n$ determine all $n\times n$ real matrices $A$ which have only real eigenvalues and such that there exists an integer $k\geq n$ with $A + A^k = A^T$ .

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 $1\leq a<b<c<d \leq n$ , the triples $\{a,b,c\}$ and $\{b,c,d\}$ have different colours. Prove that $\frac{1}{100}\log\log n \leq m \leq100\log \log n$ .

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 $p \geq 3$ be a prime number. Prove that there is a permutation $(x_1,\ldots, x_{p-1})$ of $(1,2,\ldots,p-1)$ such that $x_1x_2 + x_2x_3 + \cdots + x_{p-2}x_{p-1} \equiv 2 \pmod p$ .

Let $A_1, \ldots, A_k$ be $n\times n$ idempotent complex matrices such that $A_iA_j = -A_iA_j$ for all $1 \leq i < j \leq k$ . Prove that at least one of the matrices has rank not exceeding $\frac{n}{k}$ .

Let $n, k \geq 3$ 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 $m$ .