AoPS - Problem

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Problem

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 10.1

Tags: combinatorics

MS_Kekas

19.03.2024 20:11

Solomiya wrote the numbers $1, 2, \ldots, 2024$ on the board. In one move, she can erase any two numbers $a, b$ from the board and write the sum $a+b$ instead of each of them. After some time, all the numbers on the board became equal. What is the minimum number of moves Solomiya could make to achieve this? Proposed by Oleksiy Masalitin

CrazyInMath

19.03.2024 20:30

solve this at 1am with bad emotional state, might fakesolve

Note that when all the numbers is equal to, say,

$k$ . We have

$k\geq 2024$ . This means there would not be a number less than

$2024$ there. So every number between

$1$ and

$2023$ must be erased, which would need at least

$\lceil \frac{2023}{2}\rceil=1012$ moves. Construction is just pairing

$i$ with

$2025-i$ to make all of them

$2025$ .

ismayilzadei1387

19.03.2024 21:03

MS_Kekas wrote: Solomiya wrote the numbers $1, 2, \ldots, 2024$ on the board. In one move, she can erase any two numbers $a, b$ from the board and write the sum $a+b$ instead of each of them. After some time, all the numbers on the board became equal. What is the minimum number of moves Solomiya could make to achieve this? Proposed by Oleksiy Masalitin it is very similar to AZE Egmo tst 2021