# Solution: 2014-19 Two complex numbers

Prove that for two non-zero complex numbers $$x$$ and $$y$$, if $$|x| ,| y|\le 1$$, then $|x-y|\le |\log x-\log y|.$

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

Here is his solution of the problem 2014-19.

Alternative solutions were submitted by 박훈민 (수리과학과 2013학번, +3), 이병학 (2013학번, +3), 채석주 (2013학번, +2), 박지민 (2012학번, +3), 김범수 (2010학번, +3), 장기정 (2014학번, +3), 정경훈 (서울대 컴퓨터공학과 2006학번, +3, his solution), 윤성철 (홍익대학교 수학교육과, +3), 진형준 (인천대 2014학번, +2), 장유진 (홍익대학교 2013학번, +3), 정요한 (서울시립대학교 수학과, +3), 조현우 (경남과학고 3학년, +3).

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# Solution: 2014-15 an equation

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.)

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# Solution: 2014-18 Rank

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).

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# 2014-19 Two complex numbers

Prove that for two non-zero complex numbers $$x$$ and $$y$$, if $$|x| ,| y|\le 1$$, then $|x-y|\le |\log x-\log y|.$

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# Solution: 2014-17 Zeros of a polynomial

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).

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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 $$AB\neq 0$$.