For \( a \geq 0 \), find
\[
\lim_{n \to \infty} n \int_{-1}^0 \left( x + \frac{x^2}{2} + e^{ax} \right)^n dx.
\]

The best solution was submitted by Jang, Kijoung (장기정, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2016-02.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김동규 (수리과학과 2015학번, +3), 김동하 (기계공학과 2014학번, +3), 이상민 (수리과학과 2014학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 이종원 (수리과학과 2014학번, +3), 정성진 (수리과학과 2013학번, +3), 최대범 (2016학번, +3), 최인혁 (물리학과 2015학번, +3), Muhammaadfiruz Hasanov (2014학번, +3), 이준호 (2016학번, +2). One incorrect solution was submitted.

Prove that for every \( x_1, x_2,\ldots,x_n\in [0,1]\), there exist \(\varepsilon_1,\varepsilon_2,\ldots,\varepsilon_n\in\{1/2,-1/2\}\) such that for all \(k=1,2,\ldots,n-1\), \[ \left\lvert \sum_{i=1}^k \varepsilon_i x_i-\sum_{i=k+1}^n \varepsilon_i x_i \right\rvert\le 1.\]

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2016-1.

Alternative solutions were submitted by 노희광 (화학과 2014학번, +2), 안현수 (2016학번, +2), 이상민 (수리과학과 2014학번, +2), 홍혁표 (수리과학과 2013학번, +2). There were 10 incorrect submissions.