Let \(\theta\) be a fixed constant. Characterize all functions \(f:\mathcal R\to \mathcal R\) such that \(f”(x)\) exists for all real \(x\) and for all real \(x,y\), \[ f(y)=f(x)+(y-x)f'(x)+ \frac{(y-x)^2}{2} f”(\theta y + (1-\theta) x).\]

The best solution was submitted by 장유진 (홍익대학교 수학교육과 2013학번). Congratulations!

Here is his solution of problem 2014-15.

Alternative solutions were submitted by 박민재 (수리과학과 2011학번, +3), 장기정 (2014학번, +2), 류상우 (서울대 수리과학부 2012학번, +2), 조현우 (경남과학고 3학년, +2), 윤성철 (홍익대학교 수학교육과, +2). (The most common mistake was to assume that if a Taylor series of an infinitely differentiable function f converges, then it converges to f.)