A weird device
Source: All-Russian MO 2023 Final stage 9.8
April 23, 2023
combinatorics
Problem Statement
Petya has balls, among them there are no two balls of equal weight. He also has a device, which works as follows: if he puts exactly balls on it, it will report the sum of the weights of some two of them (but he doesn't know which ones). Prove that Petya can use the device a few times so that after a while he will be able to choose one of the balls and accurately tell its weight.