We have an expression \(x_0 \div x_1 \div x_2 \div \dots \div x_n\). A way of putting \(n-1\) left parentheses and \(n-1\) right parenthese is called ‘parenthesization’ if it is valid and completely clarify the order of operations. For example, when \(n=3\), we have the following five parenthesizations.
\[ ((x_0\div x_1)\div x_2)\div x_3, \enspace (x_0\div (x_1\div x_2))\div x_3, \enspace (x_0\div x_1)\div (x_2\div x_3),\]
\[x_0\div ((x_1\div x_2)\div x_3), \enspace x_0\div (x_1\div (x_2\div x_3)).
\]

(a) For an integer \(n\), how many parenthesizations are there?
(b) For each parenthesization, we can compute the expression to obtain a fraction with some variables in the numerator and some variable in the denominator. For an integer \(n\), determine which fraction occur most often. How many times does it occur?

The best solution was submitted by 나영준 (연세대학교 의학과 18학번, +4). Congratulations!

Here is the best solution of problem 2022-06.

Other (incomplete) solutions were submitted by 조유리 (문현여고 3학년, +2), 이명규 (KAIST 전산학부 20학번, +2), 박기찬 (KAIST 새내기과정학부 22학번, +2), Antonio Recuero Buleje (+2).

Prove or disprove the following: There exists a real \( 2 \times 2 \) matrix \( M \) such that \[
\cos M =
\begin{pmatrix}
1 & 2022 \\
0 & 1
\end{pmatrix}.
\]

The best solution was submitted by 이종민 (KAIST 물리학과 21학번, +4). Congratulations!

Here is the best solution of problem 2022-04.

Other solutions were submitted by 조유리 (문현여고 3학년, +3), 김예곤 (KAIST 수리과학과 19학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 문강연 (KAIST 새내기과정학부 22학번, +3), 박기찬 (KAIST 새내기과정학부 22학번, +3), 이호빈 (KAIST 수리과학과 대학원생, +3), 김기수 (KAIST 수리과학과 18학번, +3), 여인영 (KAIST 물리학과 20학번, +3), 권민재 (KAIST 수리과학과 19학번, +3), 채지석 (KAIST 수리과학과 대학원생, +3), 하석민 (KAIST 수리과학과 17학번, +3), 박현영 (KAIST 전자및전자공학부 대학원생, +3), 강한필 (KAIST 전산학부 16학번, +3), 이재욱 (KAIST 전기및전자공학부 대학원생, +3), 나영준 (연세대학교 의학과 18학번, +3).

For any positive integer \(n \geq 2\), let \(B_n\) be the group given by the following presentation\[ B_n = < \sigma_1, \ldots, \sigma_{n-1} | \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}, \sigma_i \sigma_j = \sigma_j \sigma_i > \]where the first relation is for \( 1 \leq i \leq n-2 \) and the second relation is for \(|i-j| \geq 2\). Show that there exists a total order < on \(B_n\) such that for any three elements \(a, b, c\in B_n\), if \(a < b\) then \(ca < cb\). 

The best solution was submitted by 박기찬 ((KAIST 새내기과정학부 22학번, +4). Congratulations!

Here is the best solution of problem 2022-02

.

For \(k,n\geq 1\), let \(v_1,\dots, v_n\) be unit vectors in \(\mathbb{R}^k\). Prove that we can always choose signs \(\varepsilon_1,\dots,\varepsilon_n\in \{-1, +1\}\) such that \(|\sum_{i=1}^{n} \varepsilon_i v_i |\leq \sqrt{n} \).

The best solution was submitted by 조유리 (문현여고 3학년, +4). Congratulations!

Here is the best solution of problem 2022-03.

Other solutions were submitted by 김예곤 (KAIST 수리과학과 19학번, +3), 구재현 (KAIST 전산학부 17학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 문강연 (KAIST 새내기과정학부 22학번, +3), 박기찬 (KAIST 새내기과정학부 22학번, +3), 이호빈 (KAIST 수리과학과 대학원생, +3), 김기수 (KAIST 수리과학과 18학번, +3), 윤창기 (서울대학교 수리과학부 19학번, +3), 권민재 (KAIST 수리과학과 19학번, +3), 유태윤 (KAIST 수리과학과 20학번, +3), 하석민 (KAIST 수리과학과 17학번, +3), 박현영 (KAIST 전자및전자공학부 대학원생, +3), 강한필 (KAIST 전산학부 16학번, +3), 여인영 (KAIST 물리학과 20학번, +2). Late solutions were not graded.

Evaluate the following:

\[ \frac{1}{1^2 \cdot 3^3 \cdot 5^2} – \frac{1}{3^2 \cdot 5^3 \cdot 7^2} + \frac{1}{5^2 \cdot 7^3 \cdot 9^2} – \dots
\]

The best solution was submitted by 여인영 (KAIST 물리학과 20학번, +4). Congratulations!

Here is the best solution of problem 2022-01.

Other solutions were submitted by 조유리 (문현여고 3학년, +3), 김건우 (KAIST 수리과학과 17학번, +3), 김예곤 (KAIST 수리과학과 19학번, +3), 신민서 (KAIST 수리과학과 20학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 조한슬 (KAIST 김재철AI대학원 대학원생, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 박기찬 (KAIST 새내기과정학부 22학번, +3), 구은한 (KAIST 수리과학과 19학번, +3), 이호빈 (KAIST 수리과학과 대학원생, +3), 김기수 (KAIST 수리과학과 18학번, +3), 이종민 (KAIST 물리학과 21학번, +2). Late solutions were not graded.

There are \(n\) people participating to a chess tournament and every two players play one game. There are no draws. Let \(a_i\) be the number of wins of the \(i\)-th player and \(b_i\) be the number of losses of the \(i\)-th player. Prove that
\[\sum_{i\in [n]} a_i^2 = \sum_{i\in [n]} b_i^2.\]

The best solution was submitted by 구재현 (전산학부 2017학번, +4). Congratulations!

Here is the best solution of problem 2021-24.

Other solutions were submitted by 이도현 (수리과학과 2018학번, +3), 이재욱 (전기및전자공학부 2018학번, +3), 이충명 (기계공학과 대학원생, +3), 이호빈 (수리과학과 대학원생, +3), 전해구 (기계공학과 졸업생, +3). Late solutions were not graded.

Let \(F\) be a family of nonempty subsets of \([n]=\{1,\dots,n\}\) such that no two disjoint subsets of \(F\) have the same union. In other words, for \(F =\{ A_1,A_2,\dots, A_k\},\) there exists no two sets \(I, J\subseteq [k]\) with \(I\cap J =\emptyset\) and \(\bigcup_{i\in I}A_i = \bigcup_{j\in J} A_j\). Determine the maximum possible size of \(F\).

For the new version of POW 2021-21, the best solution was submitted by 이재욱 (전기및전자공학부 2018학번, +4). Congratulations!

Here is the best solution of problem 2021-21.

Determine all rational numbers that can be written as
\[
\frac{1}{n_1} + \frac{1}{n_1 n_2} + \frac{1}{n_1 n_2 n_3} + \dots + \frac{1}{n_1 n_2 n_3 \dots n_k} ,
\]
where \( n_1, n_2, n_3 \dots, n_k \) are positive integers greater than \(1\).

The best solution was submitted by 조정휘 (수리과학과 대학원생, +4). Congratulations!

Here is the best solution of problem 2021-22.

Other solutions were submitted by 신주홍 (수리과학과 2020학번, +3), 이재욱 (전기및전자공학부 2018학번, +3), 이호빈 (수리과학과 대학원생, +3), 전해구 (기계공학과 졸업생, +3).

Let \(F\) be a family of nonempty subsets of \([n]=\{1,\dots,n\}\) such that no two disjoint subsets of \(F\) have the same union. Determine the maximum possible size of \(F\).

The best solution was submitted by 전해구 (기계공학과 졸업생, +4). Congratulations!

Here is the best solution of problem 2021-21.