CH-SMO 2022

Let $n$ be a positive integer. Prove that the numbers $$1^1, 3^3, 5^5, ..., (2n-1)^{2n-1}$$ all give different remainders when divided by $2^n$ .

Let $N$ be the set of positive integers. Find all functions $f : N \to N$ such that both $\bullet$ $f(f(m)f(n)) = mn$ $\bullet$ $f(2022a + 1) = 2022a + 1$ hold for all positive integers $m, n$ and $a$ .

Let $n \geq 2$ be an integer. Switzerland and Liechtenstein are performing their annual festive show. There is a field divided into $n \times n$ squares, in which the bottom-left square contains a red house with $k$ Swiss gymnasts, and the top-right square contains a blue house with $k$ Liechtensteiner gymnasts. Every other square only has enough space for a single gymnast at a time. Each second either a Swiss gymnast or a Liechtensteiner gymnast moves. The Swiss gymnasts move to either the square immediately above or to the right and the Liechtensteiner gymnasts move either to the square immediately below or to the left. The goal is to move all the Swiss gymnasts to the blue house and all the Liechtensteiner gymnasts to the red house, with the caveat that a gymnast cannot enter a house until all the gymnasts of the other nationality have left. Determine the largest $k$ in terms of $n$ for which this is possible.

For an integer $a \ge 2$ , denote by $\delta_(a)$ the second largest divisor of $a$ . Let $(a_n)_{n\ge 1}$ be a sequence of integers such that $a_1 \ge 2$ and $$a_{n+1} = a_n + \delta_(a_n)$$ for all $n \ge 1$ . Prove that there exists a positive integer $k$ such that $a_k$ is divisible by $3^{2022}$ .

Let $n\ge 3$ be an integer. Annalena has infinitely many cowbells in each of $n$ different colours. Given an integer $m \ge n + 1$ and a group of $m$ cows standing in a circle, she is tasked with tying one cowbell around the neck of every cow so that every group of $n + 1$ consecutive cows have cowbells of all the possible $n$ colours. Prove that there are only finitely many values of $m$ for which this is not possible and determine the largest such $m$ in terms of $n$ .

Let $n > 6$ be a perfect number. Let $p_1^{a_1} \cdot p_2^{a_2} \cdot ... \cdot p_k^{a_k}$ be the prime factorisation of $n$ , where we assume that $p_1 < p_2 <...< p_k$ and $a_i > 0$ for all $i = 1,...,k$ . Prove that $a_1$ is even.

Remark: An integer $n \ge 2$ is called a perfect number if the sum of its positive divisors, excluding $n$ itself, is equal to $n$ . For example, $6$ is perfect, as its positive divisors are $\{1, 2, 3, 6\}$ and $1+2+3=6$ .