IMC 2023

Find all functions $f: \mathbb{R} \to \mathbb{R}$ 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 $$A^2=B^2=C^2 \text{ and } B^3 = ABC + 2I.$$ Prove that $A^6=I$ .

Find all polynomials $P$ in two variables with real coefficients satisfying the identity $$P(x,y)P(z,t)=P(xz-yt,xt+yz).$$

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:1. he chooses a row or column of the matrix, and2. 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 $$f(\xi)+\alpha = f'(\xi)$$

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 $$W(T)=\sum_{\substack{\{u,v\}\subseteq V(T) u\neq v}}d(u,v) \text{and} H(T)=\sum_{\substack{\{u,v\}\subseteq V(T) u\neq v}}\frac{1}{d(u,v)}$$ Prove that $$W(T)\cdot H(T)\geq \frac{(n-1)^3(n+2)}{4}.$$

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 $$1+\frac{1}{1!}+\frac{1}{2!}+\dots +\frac{1}{n!}=\frac{f(n)}{g(n)}.$$ Determine whether there exists a positive integer $n$ for which $g(n)>n^{0.999n}$ .