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