# 2016-18 Partitions with equal sums

Suppose that we have a list of $$2n+1$$ integers such that whenever we remove any one of them, the remaining can be partitioned into two lists of $$n$$ integers with the same sum. Prove that all $$2n+1$$ integers are equal.

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# Solution: 2016-17 Integral with two variables

Set $L(z,w)=\int_{-2}^2\int_{-2}^2 ( \log(z-x)-\log(z-y))( \log(w-x)-\log(w-y))Q(x,y) dx dy,$
for $$z,w\in \mathbb{C}\setminus(-\infty, 2]$$, where $Q(x,y)= \frac{4-xy}{(x-y)^2\sqrt{4-x^2}\sqrt{4-y^2}}.$
Prove that $L(z,w)=2\pi^2 \log \left[ \frac{(z+R(z))(w+R(w))}{2(zw-4+R(z)R(w))} \right],$
where $$R(z)=\sqrt{z^2-4}$$ with branch cut $$[-2,2]$$.

The best solution was submitted by Choi, Daebeom (최대범, 2016학번). Congratulations!

Here is his solution of problem 2016-17. (There are a few typos.)

No alternative solutions were submitted.

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# Midterm break

The problem of the week will take a break during the midterm exam period and return on October 28, Friday. Good luck on your midterm exams!

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# Solution: 2016-16 Column spaces

Let $$A$$ be a square matrix with real entries such that $A A^T+A^T A = A+A^T.$ Prove that $$A$$ and $$A^T$$ have the same column space.

The best solution was submitted by Koon, Yun Bum (국윤범, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2016-16.

Alternative solutions were submitted by 장기정 (수리과학과 2014학번, +3), 이종원 (수리과학과 2014학번, +3), 최인혁 (물리학과 2015학번, +3), 신준형 (수리과학과 2015학번, +3), 최대범 (2016학번, +3), 김태균 (2016학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 위성군 (수리과학과 2015학번, +3), 이정환 (수리과학과 2015학번, +3). Two incorrect solutions were received.

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# 2016-17 Integral with two variables

Set $L(z,w)=\int_{-2}^2\int_{-2}^2 ( \log(z-x)-\log(z-y))( \log(w-x)-\log(w-y))Q(x,y) dx dy,$
for $$z,w\in \mathbb{C}\setminus(-\infty, 2]$$, where $Q(x,y)= \frac{4-xy}{(x-y)^2\sqrt{4-x^2}\sqrt{4-y^2}}.$
Prove that $L(z,w)=2\pi^2 \log \left[ \frac{(z+R(z))(w+R(w))}{2(zw-4+R(z)R(w))} \right],$
where $$R(z)=\sqrt{z^2-4}$$ with branch cut $$[-2,2]$$.

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Find all pairs of positive integers $$a$$ and $$b$$ such that $$a | (b^2 + b + 1)$$ and $$b | (a^2 + a + 1)$$.