# 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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