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.)

Let \(A\) and \(B\) be \(n\times n\) real matrices for an odd integer \(n\). Prove that if both \(A+A^T\) and \(B+B^T\) are invertible, then \(AB\neq 0\).

The best solution was submitted by Jimin Park (박지민, 전산학과 2012학번). Congratulations!

Here is his solution of problem 2014-18.

Alternative solutions were submitted by 채석주 (2013학번, +3), 정성진 (2013학번, +3), 장기정 (2014학번, +3), 박민재 (2011학번, +3), 김경석 (경기과학고등학교 3학년, +3).

Let \[p(x)=x^n+x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_1 x_1 + a_0\] be a polynomial. Prove that if \(p(z)=0\) for a complex number \(z\), then \[ |z| \le 1+ \sqrt{\max (|a_0|,|a_1|,|a_2|,\ldots,|a_{n-2}|)}.\]

The best solution was submitted by Minjae Park (박민재, 수리과학과 2011학번). Congratulations!

Here is his solution of the problem 2014-17.

An alternative solution was submitted by 조현우 (경남과학고등학교 3학년, +3).

For a (simple) graph \(G\), let \(o(G)\) be the number of odd-sized sets of pairwise non-adjacent vertices and let \(e(G)\) be the number of even-sized sets of pairwise non-adjacent vertices. Prove that if we can delete \(k\) vertices from \(G\) to destroy every cycle, then \[ | o(G)-e(G)|\le 2^{k}.\]

The best solution was submitted by Minjae Park (박민재, 수리과학과 2011학번). Congratulations!

Here is his solution.

An alternative solution was submitted by 김경석 (+3, 경기과학고 3학년). One incorrect solution was received (BHJ).