# 2014-15: still waiting for a good solution

We are still waiting for a good solution for Problem 2014-15.

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# Solution: 2014-16 Odd and even independent sets

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).

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# 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 $|z| \le 1+ \sqrt{\max (|a_0|,|a_1|,|a_2|,\ldots,|a_{n-2}|)}.$

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# 2014-16 Odd and even independent sets

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 $| o(G)-e(G)|\le 2^{k}.$

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# 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 $$x,y$$, $f(y)=f(x)+(y-x)f'(x)+ \frac{(y-x)^2}{2} f”(\theta y + (1-\theta) x).$

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# Solution: 2014-14 Integration and integrality

Prove or disprove that for all positive integers $$m$$ and $$n$$, $f(m,n)=\frac{2^{3(m+n)-\frac12} }{{\pi}} \int_0^{\pi/2} \sin^{ 2n – \frac12 }\theta \cdot \cos^{2m+\frac12}\theta \, d\theta$  is an integer.

The best solution was submitted by 김경석 (경기과학고등학교 3학년). Congratulations!

Here is his solution.

Alternative solutions were submitted by 이병학 (2013학번, +2), 박훈민 (2013학번, +2), 배형진 (공항중학교 3학년, +2). One incorrect solution was submitted (LSC).

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Prove or disprove that for all positive integers $$m$$ and $$n$$, $f(m,n)=\frac{2^{3(m+n)-\frac12} }{{\pi}} \int_0^{\pi/2} \sin^{ 2n – \frac12 }\theta \cdot \cos^{2m+\frac12}\theta \, d\theta$  is an integer.