IMC 2025

Let $P \in \mathbb{R}[x]$ be a polynomial with real coefficients, and suppose $\operatorname{deg}(P) \geq 2$ . 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 $\bigcup_{x \in \mathbb{R}} \ell_x=\mathbb{R}^2$ . b) Does there exist a polynomial of even degree for which the above equality still holds?

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. $A B=B A$ .

Let $a$ be an even positive integer. Find all real numbers $x$ such that $$\left\lfloor\sqrt[a]{b^a+x} \cdot b^{a-1}\right\rfloor=b^a+\lfloor x / a\rfloor$$ 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 o denotes composition. Finally, let $$f(n)=\max _{\tau \in S_n} \operatorname{ord}(\tau) \text { and } g(n)=\max _{\tau \in T_n} \operatorname{ord}(\tau) .$$ Prove that $g(n)<f(n)+n^{0.501}$ for sufficiently large $n$ .

Let $f:(0, \infty) \rightarrow \mathbb{R}$ 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 $$f(\xi)-\xi f^{\prime}(\xi)=k$$

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

For an $n \times n$ real matrix $A \in M_n(\mathbb{R})$ , denote by $A^{\mathbb{R}}$ its counter-clockwise $90^{\circ}$ rotation. (10 points) For example, $$\left[\begin{array}{lll} 1 & 2 & 3 4 & 5 & 6 7 & 8 & 9 \end{array}\right]^R=\left[\begin{array}{lll} 3 & 6 & 9 2 & 5 & 8 1 & 4 & 7 \end{array}\right]$$ Prove that if $A=A^R$ 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 $n$ sequence of $n$ distinct positive integers $X_1, X_2 \ldots, X_n$ . First, $X_1$ is chosen randomly with $\mathbb{P}\left(X_1=i\right)=2^{-i}$ for every positive integer $i$ . For $1 \leq j \leq n-1$ . 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 $\mathbb{P}\left(X_{j+1}=n_i\right)=2^{-i}$ for every positive integer $i$ . Let $Y_n=\max \left\{X_1, \ldots, X_n\right\}$ . Show that $$\mathbb{E}\left[Y_n\right]=\sum_{i=1}^n \frac{2^i}{2^i-1}$$ where $\mathbb{E}\left[Y_n\right]$ is the expected value of $Y_n$ .

For any positive integer $N$ , let $S_N$ be the number of pairs of integers $1 \leq a, b \leq N$ such that the number $\left(a^2+a\right)\left(b^2+b\right)$ is a perfect square. Prove that the limit $$\lim _{N \rightarrow \infty} \frac{S_N}{N}$$ exists and find its value.