Let \(S(n,k)\) be the Stirling number of the second kind that is the number of ways to partition a set of \(n\) objects into \(k\) non-empty subsets. Prove the following equality \[ \det\left( \begin{matrix} S(m+1,1) & S(m+1,2) & \cdots & S(m+1,n) \\
S(m+2,1) & S(m+2,2) & \cdots & S(m+2,n) \\
\cdots & \cdots & \cdots & \cdots \\
S(m+n,1) & S(m+n,2) & \cdots & S(m+n,n) \end{matrix} \right) = (n!)^m \]

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

Here is the best solution of problem 2022-15.

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 김찬우 (연세대학교 수학과, +3). Late solutions were not graded.

For a positive integer \(n\), let \(B\) and \(C\) be real-valued \(n\) by \(n\) matrices and \(O\) be the \(n\) by \(n\) zero matrix. Assume further that \(B\) is invertible and \(C\) is symmetric. Define \[A := \begin{pmatrix} O & B \\ B^T & C \end{pmatrix}.\] What is the possible number of positive eigenvalues for \(A\)?

The best solution was submitted by 김기수 (KAIST 수리과학과 18학번, +4). Congratulations!

Here is the best solution of problem 2022-14.

Prove for any \( x \geq 1 \) that

\[
\left( \sum_{n=0}^{\infty} (n+x)^{-2} \right)^2 \geq 2 \sum_{n=0}^{\infty} (n+x)^{-3}.
\]

The best solution was submitted by 김기수 (KAIST 수리과학과 18학번, +4). Congratulations!

Here is the best solution of problem 2022-13.

Another solution was submitted by 김찬우 (연세대학교 수학과, +3).

Consider the power set \(P([n])\) consisting of \(2^n\) subsets of \([n]=\{1,\dots,n\}\).
Find the smallest \(k\) such that the following holds: there exists a partition \(Q_1,\dots, Q_k\) of \(P([n])\) so that there do not exist two distinct sets \(A,B\in P([n])\) and \(i\in [k]\) with \(A,B,A\cup B, A\cap B \in Q_i\).

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

Here is the best solution of problem 2022-12.

Other solutions were submitted by 박기찬 (KAIST 새내기과정학부 22학번, +3), 김기수 (KAIST 수리과학과 18학번, +3), 신준범 (컬럼비아 대학교 20학번, +3), 이종서 (KAIST 전산학부 19학번, +3).

Does there exists a finitely generated group which contains torsion elements of order p for all prime numbers p?

The best solution was submitted by 김기수 (KAIST 수리과학과 2018학번, +4). Congratulations!

Here is the best solution of problem 2022-11

Prove or disprove the following:

For any positive integer \( n \), there exists a polynomial \( P_n \) of degree \( n^2 \) such that

(1) all coefficients of \( P_n \) are integers with absolute value at most \( n^2 \), and

(2) \( 1 \) is a root of \( P_n =0 \) with multiplicity at least \( n \).

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

Here is the best solution of problem 2022-10

Let \(A_1,\dots, A_k\) be presidential candidates in a country with \(n \geq 1\) voters with \(k\geq 2\). Candidates themselves are not voters. Each voter has her/his own preference on those \(k\) candidates.

Find maximum \(m\) such that the following scenario is possible where \(A_{k+1}\) indicates the candidate \(A_1\): for each \(i\in [k]\), there are at least \(m\) voters who prefers \(A_i\) to \(A_{i+1}\).

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

Here is the best solution of problem 2022-09.

Other solutions were submitted by 조유리 (문현여고 3학년, +3), 박기찬 (KAIST 새내기과정학부 22학번, +3), 김기수 (KAIST 수리과학과 18학번, +3), 여인영 (KAIST 물리학과 20학번, +3).

For positive integers \(n \geq 2\), let \(a_n = \lceil n/\pi \rceil \) and let \(b_n = \lceil \csc (\pi/n) \rceil \). Is \(a_n = b_n\) for all \(n \neq 3\)?

The best solution was submitted by 김기수 (KAIST 수리과학과 18학번, +4). Congratulations!

Here is the best solution of problem 2022-08.

Other solutions were submitted by 조유리 (문현여고 3학년, +3), 이명규 (KAIST 전산학부 20학번, +3), 박기찬 (KAIST 새내기과정학부 22학번, +3), 여인영 (KAIST 물리학과 20학번, +3).

Prove the following identity for \( x, y \in \mathbb{R}^3 \):
\[
\frac{1}{|x-y|} = \frac{1}{\pi^3} \int_{\mathbb{R}^3} \frac{1}{|x-z|^2} \frac{1}{|y-z|^2} dz.
\]

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

Here is the best solution of problem 2022-07.

Other solutions were submitted by 이종민 (KAIST 물리학과 21학번, +3), 박기찬 (KAIST 새내기과정학부 22학번, +3), 김기수 (KAIST 수리과학과 18학번, +3), 여인영 (KAIST 물리학과 20학번, +3), 채지석 (KAIST 수리과학과 대학원생, +3), 이상민 (KAIST 수리과학과 대학원생, +3), 나영준 (연세대학교 의학과 18학번, +3).

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).