IMC 2023

Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that have a continuous second derivative and for which the equality $f(7x + 1) = 49f(x)$ holds for all $x \in \mathbb{R}$.

Let $A$, $B$ and $C$ be $n \times n$ matrices with complex entries satisfying

Prove that $A^6 = I$.

Find all polynomials $P$ in two variables with real coefficients satisfying the identity

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

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 $x = (x_1, x_2, \ldots, x_n)$ such that $\{x_1, \ldots, x_n\} = M$. Ivan prefers some (at least one) of these permutations. He realized that for every preferred permutation $x$, there exist $k$ indices $i_1 < i_2 < \ldots < i_k$ with the following property: for every $1 \le j < k$, if he swaps $x_{i_j}$ and $x_{i_{j+1}}$, he obtains another preferred permutation.

Prove that he prefers at least $k!$ permutations.

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 multiplies or divides the chosen row or column entry-wise by the other row or column, respectively.

Can Ivan end up with the matrix $\begin{pmatrix} 2 & 4 \\ 2 & 3 \end{pmatrix}$ after finitely many steps?

Let $V$ be the set of all continuous functions $f \colon [0, 1] \to \mathbb{R}$, 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 some $\xi \in (0, 1)$ such that

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 shortest path in $T$ that connects $u$ with $v$.

Consider the sums

Prove that

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 two of the three coordinate axes), the projections of $X$ and $Y$ onto that plane are disjoint.

Find $\sup\{V \mid V \text{ is good}\}$.

For every positive integer $n$, let $f(n), g(n)$ be the minimal positive integers such that

Determine whether there exists a positive integer $n$ for which $g(n) > n^{0.999n}$.