# Solution: 2022-12 A partition of the power set of a set

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!

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).

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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 $$k,n\geq 1$$, let $$v_1,\dots, v_n$$ be unit vectors in $$\mathbb{R}^k$$. Prove that we can always choose signs $$\varepsilon_1,\dots,\varepsilon_n\in \{-1, +1\}$$ such that $$|\sum_{i=1}^{n} \varepsilon_i v_i |\leq \sqrt{n}$$.