2020-13 An integral sequence

Let \( a_n \) be a sequence defined recursively by \( a_0 = a_1 = \dots = a_5 = 1 \) and
\[
a_n = \frac{a_{n-1} a_{n-5} + a_{n-2} a_{n-4} + a_{n-3}^2}{a_{n-6}}
\]
for \( n \geq 6 \). Prove or disprove that every \( a_n \) is an integer.

Solution: 2020-12 Draws on a chess tournament

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.

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2020-12.

Another solution was submitted by 고성훈 (수리과학과 2018학번, +3).

Solution: 2020-11 Free group of rank 2

Show that there is a subgroup of a free group of ran 2 that is not finitely generated.

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2020-11.

Other solutions were submitted by 조한슬 (수리과학과 2017학번, +3), 최백규(생명과학과 대학원, +2).

2020-12 Draws on a chess tournament

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.

Notice on POW 2020-11

POW 2020-11 is still open and anyone who first submits a correct solution will get the full credit.

2020-11 Free group of rank 2

Show that there is a subgroup of a free group of ran 2 that is not finitely generated.

Solution: 2020-10 An inequality with sin and log

Prove that
\[\frac{x+\sin x}{2} \geq \log (1+x)\]
for \( x > -1 \).

An incomplete solution was submitted by 유찬진 (수리과학과 2015학번, +2).

2020-10 An inequality with sin and log

Prove that
\[
\frac{x+\sin x}{2} \geq \log (1+x)
\]
for \( x > -1 \).

Solution: 2020-09 Displacement of permutations

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.