IMC 2025

Let $P \in \mathbb{R}[x]$ be a polynomial with real coefficients, and suppose $\deg(P) \ge 2$. For every $x \in \mathbb{R}$, let $\ell_x \subset \mathbb{R}^2$ denote the line tangent to the graph of $P$ at the point $(x, P(x))$.

(a) Suppose that the degree of $P$ is odd. Show that $\displaystyle\bigcup_{x \in \mathbb{R}} \ell_x = \mathbb{R}^2$.

(b) Does there exist a polynomial of even degree for which the above equality still holds?

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a twice continuously differentiable function, and suppose that $\int_{-1}^{1} f(x)\,\mathrm{d}x = 0$ and $f(1) = f(-1) = 1$. Prove that

and find all such functions for which equality holds.

Denote by $\mathcal{S}$ the set of all real symmetric $2025 \times 2025$ matrices of rank $1$ whose entries take values $-1$ or $+1$. Let $A, B \in \mathcal{S}$ be matrices chosen independently uniformly at random. Find the probability that $A$ and $B$ commute, i.e. $AB = BA$.

Let $a$ be an even positive integer. Find all real numbers $x$ such that

holds for every positive integer $b$.

(Here $\lfloor x \rfloor$ denotes the largest integer that is no greater than $x$.)

For a positive integer $n$, let $[n] = \{1, 2, \ldots, n\}$. Denote by $S_n$ the set of all bijections from $[n]$ to $[n]$, and let $T_n$ be the set of all maps from $[n]$ to $[n]$. Define the order $\operatorname{ord}(\tau)$ of a map $\tau \in T_n$ as the number of distinct maps in the set $\{\tau, \tau \circ \tau, \tau \circ \tau \circ \tau, \ldots\}$ where $\circ$ denotes composition. Finally, let

Prove that $g(n) < f(n) + n^{0.501}$ for sufficiently large $n$.

Let $f \colon (0, \infty) \to \mathbb{R}$ be a continuously differentiable function, and let $b > a > 0$ be real numbers such that $f(a) = f(b) = k$. Prove that there exists a point $\xi \in (a, b)$ such that

Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all nonempty subsets $M \subseteq \mathbb{Z}_{>0}$ satisfying both of the following properties:

(a) if $x \in M$, then $2x \in M$,

(b) if $x, y \in M$ and $x + y$ is even, then $\dfrac{x+y}{2} \in M$.

For an $n \times n$ real matrix $A \in M_n(\mathbb{R})$, denote by $A^{\mathsf{R}}$ its counter-clockwise $90°$ rotation. For example,

Prove that if $A = A^{\mathsf{R}}$ then for any eigenvalue $\lambda$ of $A$, we have $\operatorname{Re}\lambda = 0$ or $\operatorname{Im}\lambda = 0$.

Let $n$ be a positive integer. Consider the following random process which produces a sequence of $n$ distinct positive integers $X_1, X_2, \ldots, X_n$.

First, $X_1$ is chosen randomly with $\mathbb{P}(X_1 = i) = 2^{-i}$ for every positive integer $i$. For $1 \le j \le n-1$, having chosen $X_1, \ldots, X_j$, arrange the remaining positive integers in increasing order as $n_1 < n_2 < \cdots$, and choose $X_{j+1}$ randomly with $\mathbb{P}(X_{j+1} = n_i) = 2^{-i}$ for every positive integer $i$.

Let $Y_n = \max\{X_1, \ldots, X_n\}$. Show that

where $\mathbb{E}[Y_n]$ is the expected value of $Y_n$.

For any positive integer $N$, let $S_N$ be the number of pairs of integers $1 \le a, b \le N$ such that the number $(a^2 + a)(b^2 + b)$ is a perfect square. Prove that the limit

exists and find its value.