Problem catalog

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) …

Let $f : \mathbb{R} \to \mathbb{R}$ be a twice continuously differentiable function, and suppose that $\int_{-1}^{1} f(x)\,dx = 0$ and $f(1) = f(-1) = 1$. Prove that $\int_{-1}^{1} (f''(x))^2\,dx \ge 15,$ and find all such functions for …

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

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 $\lfloor x \rfloor$ …

For a positive integer $n$, let $[n] = \{1, 2, \dots, n\}$. 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 …

Let $f : (0, \infty) \to \mathbb{R}$ 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 $f(\xi) - \xi f'(\xi) = k.$ …

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 $2x \in M$, (b) if $x, y \in M$ and $x + y$ is even, then …

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

Let $n$ be a positive integer. Consider the following random process which produces a sequence of $n$ distinct positive integers $X_1, X_2, \dots, X_n$. First, $X_1$ is chosen randomly with $\mathsf{P}(X_1 = i) = 2^{-i}$ for every positive …

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 $\lim_{N \to \infty} \frac{S_N}{N}$ exists and find its …

Determine all pairs $(a, b) \in \mathbb{C} \times \mathbb{C}$ satisfying $|a| = |b| = 1 \quad \text{and} \quad a + b + a\bar{b} \in \mathbb{R}.$ (proposed by Mike Daas, Universiteit Leiden)

For $n = 1, 2, \dots$ let $S_n = \log \Bigl( \sqrt[n^2]{1^1 \cdot 2^2 \cdot \dots \cdot n^n} \Bigr) - \log(\sqrt{n}),$ where $\log$ denotes the natural logarithm. Find $\lim\limits_{n \to \infty} S_n$. (proposed by Sergey Chernov, …

For which positive integers $n$ does there exist an $n \times n$ matrix $A$ whose entries are all in $\{0, 1\}$, such that $A^2$ is the matrix of all ones? (proposed by Alex Avdiushenko, Neapolis University Paphos, Cyprus)

Let $g$ and $h$ be two distinct elements of a group $G$, and let $n$ be a positive integer. Consider a sequence $w = (w_1, w_2, \dots)$ which is not eventually periodic and where each $w_i$ is either $g$ or $h$. Denote by $H$ the subgroup …

Let $n > d$ be positive integers. Choose $n$ independent, uniformly distributed random points $x_1, \dots, x_n$ in the unit ball $B \subset \mathbb{R}^d$ centered at the origin. For a point $p \in B$ denote by $f(p)$ the probability that …

Prove that for any function $f : \mathbb{Q} \to \mathbb{Z}$, there exist $a, b, c \in \mathbb{Q}$ such that $a < b < c$, $f(b) \ge f(a)$, and $f(b) \ge f(c)$. (proposed by Mehdi Golafshan &amp; Markus A. Whiteland, University of Liège, …

Let $n$ be a positive integer. Suppose that $A$ and $B$ are invertible $n \times n$ matrices with complex entries such that $A + B = I$ (where $I$ is the identity matrix) and $(A^2 + B^2)(A^4 + B^4) = A^5 + B^5.$ Find all possible values …

Define the sequence $x_1, x_2, \dots$ by the initial terms $x_1 = 2$, $x_2 = 4$, and the recurrence relation $x_{n+2} = 3 x_{n+1} - 2 x_n + \frac{2^n}{x_n} \quad \text{for } n \ge 1.$ Prove that …

A matrix $A = (a_{ij})$ is called nice, if it has the following properties: (i) the set of all entries of $A$ is $\{1, 2, \dots, 2t\}$ for some integer $t$; (ii) the entries are non-decreasing in every row and in every column: …

We say that a square-free positive integer $n$ is almost prime if $n \mid x^{d_1} + x^{d_2} + \dots + x^{d_k} - kx$ for all integers $x$, where $1 = d_1 < d_2 < \dots < d_k = n$ are all the positive divisors of $n$. Suppose that $r$ …

Find all functions $f : \mathbb{R} \to \mathbb{R}$ that have a continuous second derivative and for which the equality $f(7x + 1) = 49 f(x)$ holds for all $x \in \mathbb{R}$. (proposed by Alex Avdiushenko, Neapolis University Paphos, …

Let $A$, $B$ and $C$ be $n \times n$ matrices with complex entries satisfying $A^2 = B^2 = C^2 \quad \text{and} \quad B^3 = ABC + 2I.$ Prove that $A^6 = I$. (proposed by Mike Daas, Universiteit Leiden)

Find all polynomials $P$ in two variables with real coefficients satisfying the identity $P(x, y) P(z, t) = P(xz - yt, xt + yz).$ (proposed by Giorgi Arabidze, Free University of Tbilisi, Georgia)

Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_i = i^k + i$ for $i = 0, 1, \dots, p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_2$ upon division by …

Fix positive integers $n$ and $k$ such that $2 \le k \le n$ and a set $M$ consisting of $n$ fruits. A permutation is a sequence $x = (x_1, x_2, \dots, x_n)$ such that $\{x_1, \dots, x_n\} = M$. Ivan prefers some (at least …

Ivan writes the matrix $\begin{pmatrix} 2 & 3 \\ 2 & 4 \end{pmatrix}$ on the board. Then he performs the following operation on the matrix several times:

• he chooses a row or a column of the matrix, and
• he …

Let $V$ be the set of all continuous functions $f : [0, 1] \to \mathbb{R}$, differentiable on $(0, 1)$, with the property that $f(0) = 0$ and $f(1) = 1$. Determine all $\alpha \in \mathbb{R}$ such that for every $f \in V$, there exists …

Let $T$ be a tree with $n$ vertices; that is, a connected simple graph on $n$ vertices that contains no cycle. For every pair $u, v$ of vertices, let $d(u, v)$ denote the distance between $u$ and $v$, that is, the number of edges in the …

We say that a real number $V$ is good if there exist two closed convex subsets $X$, $Y$ of the unit cube in $\mathbb{R}^3$, with volume $V$ each, such that for each of the three coordinate planes (that is, the planes spanned by any …

For every positive integer $n$, let $f(n)$, $g(n)$ be the minimal positive integers such that $1 + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} = \frac{f(n)}{g(n)}.$ Determine whether there exists a positive integer $n$ for which …