Problem

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 11.3

Tags: algebra, combinatorics, poset



Let's define almost mean of numbers $a_1, a_2, \ldots, a_n$ as $\frac{a_1 + a_2 + \ldots + a_n}{n+1}$. Oleksiy has positive real numbers $b_1, b_2, \ldots, b_{2023}$, not necessarily distinct. For each pair $(i, j)$ with $1 \leq i, j \leq 2023$, Oleksiy wrote on a board almost mean of numbers $b_i, b_{i+1}, \ldots, b_j$. Prove that there are at least $45$ distinct numbers on the board. Proposed by Anton Trygub