There are \(n\) people participating to a chess tournament and every two players play exactly one game against each other. The winner receives \(1\) point and the loser gets \(0\) point and if the game is a draw, each player receives \(0.5\) points. Prove that if at least \(3/4\) of the games are draws, then there are two players with the same total scores.
POW 2020-11 is still open and anyone who first submits a correct solution will get the full credit.
Show that there is a subgroup of a free group of ran 2 that is not finitely generated.
Prove that
\[\frac{x+\sin x}{2} \geq \log (1+x)\]
for \( x > -1 \).
An incomplete solution was submitted by 유찬진 (수리과학과 2015학번, +2).
Prove that
\[
\frac{x+\sin x}{2} \geq \log (1+x)
\]
for \( x > -1 \).
For a permutation \(\pi: [n]\rightarrow [n]\), we define the displacement of \(\pi\) to be \(\sum_{i\in [n]} |i-\pi(i)|\).
For given \(k\), prove that the number of even permutations of \([n]\) with displacement \(2k\) minus the number of odd permutations of \([n]\) with displacement \(2k\) is \((-1)^{k}\binom{n-1}{k}\).
The best solution was submitted by 홍의천 (수리과학과 2017학번). Congratulations!
Here is his solution of problem 2020-09.
Another solution was submitted by 고성훈 (수리과학과 2018학번, +3).
In the problem 2019-08 (https://mathsci.kaist.ac.kr/pow/2019/2019-08-group-action/), we considered a group G acting by isometries on a proper geodesic metric space X properly discontinuously and cocompactly. Such an action is called a geometric action. The conclusion was that a geometric action leads to that G is finitely generated.
Would this conclusion still hold in the case the space X is not necessarily proper?
The best solution was submitted by 홍의천 (수리과학과 2017학번). Congratulations!
Here is his solution of problem 2020-08.
For a permutation \(\pi: [n]\rightarrow [n]\), we define the displacement of \(\pi\) to be \(\sum_{i\in [n]} |i-\pi(i)|\).
For given \(k\), prove that the number of even permutations of \([n]\) with displacement \(2k\) minus the number of odd permutations of \([n]\) with displacement \(2k\) is \((-1)^{k}\binom{n-1}{k}\).
Due to a technical issue, POW 2020-08 was posted on Monday. Correspondingly, the due date for POW 2020-08 is postponed to May 29, Sat. (by noon).
In the problem 2019-08 (https://mathsci.kaist.ac.kr/pow/2019/2019-08-group-action/), we considered a group G acting by isometries on a proper geodesic metric space X properly discontinuously and cocompactly. Such an action is called a geometric action. The conclusion was that a geometric action leads to that G is finitely generated.
Would this conclusion still hold in the case the space X is not necessarily proper?