IMC 2021

Let $A$ be a real $n \times n$ matrix such that $A^3 = 0$.

(a) Prove that there is a unique real $n \times n$ matrix $X$ that satisfies the equation

(b) Express $X$ in terms of $A$.

Let $n$ and $k$ be fixed positive integers, and let $a$ be an arbitrary non-negative integer. Choose a random $k$-element subset $X$ of $\{1, 2, \ldots, k + a\}$ uniformly (i.e., all $k$-element subsets are chosen with the same probability) and, independently of $X$, choose a random $n$-element subset $Y$ of $\{1, \ldots, k + n + a\}$ uniformly.

Prove that the probability

does not depend on $a$.

We say that a positive real number $d$ is good if there exists an infinite sequence $a_1, a_2, a_3, \ldots \in (0, d)$ such that for each $n$, the points $a_1, \ldots, a_n$ partition the interval $[0, d]$ into segments of length at most $1/n$ each. Find

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a function. Suppose that for every $\varepsilon > 0$, there exists a function $g \colon \mathbb{R} \to (0, \infty)$ such that for every pair $(x, y)$ of real numbers,

Prove that $f$ is the pointwise limit of a sequence of continuous $\mathbb{R} \to \mathbb{R}$ functions, i.e., there is a sequence $h_1, h_2, \ldots$ of continuous $\mathbb{R} \to \mathbb{R}$ functions such that $\lim\limits_{n \to \infty} h_n(x) = f(x)$ for every $x \in \mathbb{R}$.

Let $A$ be a real $n \times n$ matrix and suppose that for every positive integer $m$ there exists a real symmetric matrix $B$ such that

Prove that $|\det A| \le 1$.

For a prime number $p$, let $\mathrm{GL}_2(\mathbb{Z}/p\mathbb{Z})$ be the group of invertible $2 \times 2$ matrices of residues modulo $p$, and let $S_p$ be the symmetric group (the group of all permutations) on $p$ elements. Show that there is no injective group homomorphism $\varphi \colon \mathrm{GL}_2(\mathbb{Z}/p\mathbb{Z}) \to S_p$.

Let $D \subseteq \mathbb{C}$ be an open set containing the closed unit disk $\{z : |z| \le 1\}$. Let $f \colon D \to \mathbb{C}$ be a holomorphic function, and let $p(z)$ be a monic polynomial. Prove that

Let $n$ be a positive integer. At most how many distinct unit vectors can be selected in $\mathbb{R}^n$ such that from any three of them, at least two are orthogonal?