Question 162
A group of 4050 friends plays a video game tournament. For this, there are 2025 computers in one room labelled $a_1, \dots, a_{2025}$ and 2025 computers in another room labelled $b_1, \dots, b_{2025}$. The player at computer $a_i$ always plays against the players $b_i$, $b_{i+2}$, $b_{i+3}$ and $b_{i+4}$ (so in particular not against $b_{i+1}$), where the computers are numbered cyclically. After the first round, all players choose a computer within their own room for the second round. Afterwards they notice that everyone has the same opponents in the second round as in the first round.
Prove that if someone chose the same computer in both rounds, then everyone chose the same computer in both rounds.
Prove that if someone chose the same computer in both rounds, then everyone chose the same computer in both rounds.