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).