Problem

Source: 2022 Saudi Arabia January Camp Test 2.3 BMO + EGMO TST

Tags: combinatorics, number theory



Let $n$ be an even positive integer. On a board n real numbers are written. In a single move we can erase any two numbers from the board and replace each of them with their product. Prove that for every $n$ initial numbers one can in finite number of moves obtain $n$ equal numbers on the board.