Prove that for each positive integer \(n\), there exist \(n\) real numbers \(x_1,x_2,\ldots,x_n\) such that \[\sum_{j=1}^n \frac{x_j}{1-4(i-j)^2}=1 \text{ for all }i=1,2,\ldots,n\] and \[\sum_{j=1}^n x_j=\binom{n+1}{2}.\]
The best solution was submitted by Taehyun Eom (엄태현), 2012학번. Congratulations!
Here is his Solution of Problem 2012-23.
Alternative solutions were submitted by 박민재 (2011학번, +3, Solution), 김태호 (수리과학과 2011학번, +2), 이명재 (2012학번, +2).
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http://mjlee93.tistory.com/attachment/cfile25.uf@170BCA3650BD70B92FF3BF.pdf