# 2022-07 Coulomb potential

Prove the following identity for $$x, y \in \mathbb{R}^3$$:
$\frac{1}{|x-y|} = \frac{1}{\pi^3} \int_{\mathbb{R}^3} \frac{1}{|x-z|^2} \frac{1}{|y-z|^2} dz.$

Solutions are due May 6th (Friday), 6PM, and it will remain open if nobody solved it.

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# Solution: 2022-06 A way of putting parentheses

We have an expression $$x_0 \div x_1 \div x_2 \div \dots \div x_n$$. A way of putting $$n-1$$ left parentheses and $$n-1$$ right parenthese is called ‘parenthesization’ if it is valid and completely clarify the order of operations. For example, when $$n=3$$, we have the following five parenthesizations.
$((x_0\div x_1)\div x_2)\div x_3, \enspace (x_0\div (x_1\div x_2))\div x_3, \enspace (x_0\div x_1)\div (x_2\div x_3),$
$x_0\div ((x_1\div x_2)\div x_3), \enspace x_0\div (x_1\div (x_2\div x_3)).$

(a) For an integer $$n$$, how many parenthesizations are there?
(b) For each parenthesization, we can compute the expression to obtain a fraction with some variables in the numerator and some variable in the denominator. For an integer $$n$$, determine which fraction occur most often. How many times does it occur?

The best solution was submitted by 나영준 (연세대학교 의학과 18학번, +4). Congratulations!

Other (incomplete) solutions were submitted by 조유리 (문현여고 3학년, +2), 이명규 (KAIST 전산학부 20학번, +2), 박기찬 (KAIST 새내기과정학부 22학번, +2), Antonio Recuero Buleje (+2).

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# Notice: Mid-term break

POW will resume on Apr. 29.

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# Solution: 2022-05 squares of perfect squares

Show that there do not exist perfect squares a, b, c such that $$a^2 + b^2 = c^2$$, provided that a, b, c are nonzero integers.

The best solution was submitted by 박준성 (KAIST 수리과학과 19학번, +4). Congratulations!

Other solutions were submitted by 조유리 (문현여고 3학년, +3), 김예곤 (KAIST 수리과학과 19학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 문강연 (KAIST 새내기과정학부 22학번, +3), 박기찬 (KAIST 새내기과정학부 22학번, +3), 이호빈 (KAIST 수리과학과 대학원생, +3), 김기수 (KAIST 수리과학과 18학번, +3), 이재욱 (KAIST 전기및전자공학부 대학원생, +3).

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# 2022-06 A way of putting parentheses

We have an expression $$x_0 \div x_1 \div x_2 \div \dots \div x_n$$. A way of putting $$n-1$$ left parentheses and $$n-1$$ right parenthese is called ‘parenthesization’ if it is valid and completely clarify the order of operations. For example, when $$n=3$$, we have the following five parenthesizations.
$((x_0\div x_1)\div x_2)\div x_3, \enspace (x_0\div (x_1\div x_2))\div x_3, \enspace (x_0\div x_1)\div (x_2\div x_3),$
$x_0\div ((x_1\div x_2)\div x_3), \enspace x_0\div (x_1\div (x_2\div x_3)).$

(a) For an integer $$n$$, how many parenthesizations are there?
(b) For each parenthesization, we can compute the expression to obtain a fraction with some variables in the numerator and some variable in the denominator. For an integer $$n$$, determine which fraction occur most often. How many times does it occur?

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# Notice on POW 2022-05 (Problem Revision)

I hope you noticed the day this problem appeared was April fool’s day. However, we sincerely apologize to the students got confused about the problem description, and we found that many students already submitted the solution corresponding to the original problem.

Hence We revise the problem as the following:

Show that there do not exist perfect squares a, b, c such that $$a^2 + b^2 = c^2$$, provided that a, b, c are nonzero integers.

You should actually provide the full valid proof i.e. the solution like ‘It is the special case of some famous theorem hence it is trivial’ will not be graded. Please resubmit your solution if you already submitted the solution for the previous version.

We accept the solution until April 11 Monday, 6PM.

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# Solution: 2022-04 Cosine matrix

Prove or disprove the following: There exists a real $$2 \times 2$$ matrix $$M$$ such that $\cos M = \begin{pmatrix} 1 & 2022 \\ 0 & 1 \end{pmatrix}.$

The best solution was submitted by 이종민 (KAIST 물리학과 21학번, +4). Congratulations!

Other solutions were submitted by 조유리 (문현여고 3학년, +3), 김예곤 (KAIST 수리과학과 19학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 문강연 (KAIST 새내기과정학부 22학번, +3), 박기찬 (KAIST 새내기과정학부 22학번, +3), 이호빈 (KAIST 수리과학과 대학원생, +3), 김기수 (KAIST 수리과학과 18학번, +3), 여인영 (KAIST 물리학과 20학번, +3), 권민재 (KAIST 수리과학과 19학번, +3), 채지석 (KAIST 수리과학과 대학원생, +3), 하석민 (KAIST 수리과학과 17학번, +3), 박현영 (KAIST 전자및전자공학부 대학원생, +3), 강한필 (KAIST 전산학부 16학번, +3), 이재욱 (KAIST 전기및전자공학부 대학원생, +3), 나영준 (연세대학교 의학과 18학번, +3).

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Show that there exist perfect squares a, b, c such that $$a^2 + b^2 = c^2$$.
I hope you noticed the day this problem appeared was April fool’s day. Show instead that there do not exist perfect squares a, b, c such that $$a^2 + b^2 = c^2$$, provided that a, b, c are nonzero integers.