In addition to other answers, one way to think about it is that the options are symmetric around the midpoint: a guess that partitions the space into (1/4 of options, 3/4 of options) is the same as one that does (3/4, 1/4). So (1/2, 1/2) is special in some way – it has to be either a local minimum or local maximum. And if the function is convex (or close enough), then (1/2, 1/2) is a global minimum/maximum.
But (1/2, 1/2) is clearly a better choice than just guessing a specific individual. So it must be the best choice.
This is all right, but it just kicks the intuition into the assumption that the function is convex. As far as I can tell from the paper, this turns out to be exactly the argument they use to prove that (1/2, 1/2) is the optimal guess. But the majority of that proof is dedicated to showing that the function is indeed convex.
In the IEEEXTREME university programming competition there are ~10k participating teams.
Our university has a quite strong Competitive Programming program and the best teams usually rank in the top 100. Last year a team ranked 30 and it's wasn't even our strongest team (which didn't participate)
This year none of our teams was able to get in the top 1000. I would estimate close to 99% of the teams in the Top 1000 were using LLMs.
Last year they didn't seem to help much, but this year they rendered the competition pointless.
I've read blogs/seen videos of people who got in the AOC global leaderboard last year without using LLMs, but I think this year it wouldn't be possible at all.