UA-UMO 2023 Problem 82

In one country, a one-round tennis tournament was held (everyone played with everyone exactly once). Participants received $1$ point for winning a match, and $0$ points for losing. There are no draws in tennis. At the end of the tournament, Oleksiy saw the number of points scored by each participant, as well as the schedule of all the matches in the tournament, which showed the pairs of players, but not the winners. He chooses matches one by one in any order he wants and tries to guess the winner, after which he is told if he is correct. Prove that Oleksiy can act in such a way that he is guaranteed to guess the winners of more than half of the matches.

Proposed by Oleksiy Masalitin