Let \(f(x)\in \mathbb R[x]\) be a polynomial of degree at most \(n\) such that \[ x^2+f(x)^2\le 1\] for all \( -1\le x\le 1 \). Prove that \( \lvert f'(x)\rvert \le 2(n-1)\) for all \( -1\le x\le 1\).
The best solution was submitted by Huy Tùng Nguyễn (수리과학과 2016학번). Congratulations!
Here is his solution of problem 2017-14.
Alternative solutions were submitted by 유찬진 (수리과학과 2015학번, +3), 이본우 (2017학번, +2). One incorrect solution was submitted.