# Solution: 2015-8 all lines

Does there exist a subset $$A$$ of $$\mathbb{R}^2$$ such that $$\lvert A\cap L\rvert=2$$ for every straight line $$L$$?

The best solution was submitted by Lee, Su Cheol (이수철, 수리과학과 2012학번). Congratulations!

Here is his solution of problem 2015-08.

Alternative solutions were submitted by 김기현 (수리과학과 2012학번, +3), 김동률 (2015학번, +3), 엄태현 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 진우영 (수리과학과 2012학번, +3), 박훈민 (수리과학과 2013학번, +2), 오동우 (2015학번, +2).

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Let $$n\ge 1$$ and $$a_0,a_1,a_2,\ldots,a_{n}$$ be non-negative integers. Prove that if $N=\frac{a_0^2+a_1^2+a_2^2+\cdots+a_{n}^2}{1+a_0a_1a_2\cdots a_{n}}$ is an integer, then $$N$$ is the sum of $$n$$ squares of integers.