# Notice on POW 2022-10

POW 2022-10 is still open. Anyone who first submits a correct solution will get the full credit.

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# 2022-11 groups with torsions

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

Solutions for POW 2022-11 are due July 4th (Saturday), 12PM, and it will remain open if nobody solved it.

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# Solution: 2022-09 A chaotic election

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!

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

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# 2022-10 Polynomial with root 1

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 $$n$$.

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# Solution: 2022-08 two sequences

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!

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

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# 2022-09 A chaotic election

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 $$A_{i+1}$$.

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# Solution: 2022-07 Coulomb potential

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!

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).

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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 $$n \neq 3$$?