10 Chinese and 10 Japanese people are sitting in a round table that can rotate both sides. On the table, there are 20 personal screens with a movie on them, 10 movies in Chinese and 10 in Japanese. Prove that for any arrangement of people and screens, we can rotate the table(screens) in such a way there will be at least 10 people with the right screen (a screen with a language they can understand).
First, I noticed that if 5 or more Japanese people have the right screen, then at least 5 Chinese and the opposite, so we need to prove that we can arrange the screens so that at least 5 of any nationality would get the right screen.
with a small amount of people like 2 3 or 4 of each nationality, it is very clear but for 5 or more people it's harder
Consider any original starting setup, and for the following argument, let's fix one person (of either ethnicity):
If we consider all $20$ possible rotations, this person will clearly have the correct language film on their screen exactly $10$ times. This extends to every person at the table (as the person was picked arbitrarily), and so this means that over all $20$ possible rotations, we have $20\cdot10 = 200$ total correctly assigned films.
But then this means that regardless of original arrangement, over all $20$ possible rotations, there is a mean of $200/20=10$ people with screens in the correct language. This implies that out of all $20$ possible rotations, either all arrangements have exactly $10$ people with correct screens, or that we have at least one rotation with fewer than $10$ correct assignments, however this immediately implies that we have at least one rotation with more than $10$ correct assignments (otherwise the mean would have to be below $10$). In either case, we have found at least one rotation with at least $10$ correctly assigned screens.