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

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

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