# Solution: 2018-14 Forests and Planes

Suppose that the edges of a graph $$G$$ can be colored by 3 colors so that there is no monochromatic cycle. Prove or disprove that $$G$$ has two planar subgraphs $$G_1,G_2$$ such that $$E(G)=E(G_1)\cup E(G_2)$$.

The best solution was submitted by Huy Tùng Nguyễn (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2018-14.

Alternative solutions were submitted by 강한필 (전산학부 2016학번, +3, solution) and 김일희 (수리과학과 2001학번 동문, +3, solution). There was one incorrect submission.

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# Solution: 2017-22 Debugging

Let $$p$$, $$q$$, $$r$$ be positive integers such that $$p,q\ge r$$. Ada and Betty independently read all source codes of their programming project. Ada found $$p$$ bugs and Betty found $$q$$ bugs, including $$r$$ bugs that Ada found. What is the expected number of remaining bugs that neither Ada nor Betty found?

The best solution was submitted by Huy Tùng Nguyễn (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2017-22.

Alternative solutions were submitted by 최대범 (수리과학과 2016학번, +3), 김태균 (수리과학과 2016학번, +2), 유찬진 (수리과학과 2015학번, +2). One incorrect solution was received.

(This is the last problem of this semester. Thank you all for participating POW.)

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# Solution: 2017-14 Polynomials of degree at most n

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.

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# Concluding 2017 Spring

Thanks all for participating POW actively. Here’s the list of winners:

1st prize (Gold): Jo, Tae Hyouk (조태혁, 수리과학과 2014학번)
2nd prize (Silver): Huy Tùng Nguyễn (수리과학과 2016학번)
2nd prize (Silver): 최대범 (수리과학과 2016학번)
2nd prize (Silver): Lee, Bonwoo (이본우, 2017학번)
3rd prize (Bronze): Jang, Kijoung (장기정, 수리과학과 2014학번)

조태혁 (수리과학과 2014학번) 36/40
Huy Tung Nguyen (2016학번) 35/40
최대범 (수리과학과 2016학번) 31/40
이본우 (2017학번) 30/40
장기정 (수리과학과 2014학번) 26/40
위성군 (수리과학과 2015학번) 25/40
최인혁 (물리학과 2015학번) 25/40
오동우 (수리과학과 2015학번) 24/40
김태균 (수리과학과 2016학번) 20/40
Ivan Adrian Koswara (전산학부 2013학번) 12/40
강한필 (2016학번) 9/40
유찬진 (수리과학과 2015학번) 4/40
채지석 (2016학번) 3/40
곽상훈 (수리과학과 2013학번) 3/40
김재현 (수리과학과 2016학번) 3/40
이정환 (수리과학과 2015학번) 3/40
이준호 (2016학번) 3/40
홍혁표 (수리과학과 2013학번) 3/40
이태영 (수리과학과 2013학번) 2/40

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# Solution: 2017-09 A Diophantine Equation

Find all positive integers $$a, b, c$$ satisfying $3^a + 5^b = 2^c.$

The best solution was submitted by Huy Tùng Nguyễn (2016학번). Congratulations!

Here is his solution of problem 2017-09.

Alternative solutions were submitted by 조태혁 (수리과학과 2014학번, +3), 이본우 (2017학번, +3), 최대범 (수리과학과 2016학번, +2), 이재우 (함양고등학교 2학년, +2).

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# Solution: 2017-08 Long arithmetic progression

Does there exist a constant $$\varepsilon>0$$ such that for each positive integer $$n$$ and each subset $$A$$ of $$\{1,2,\ldots,n\}$$ with $$\lvert A\rvert<\varepsilon n$$, there exists an artihmetic progression $$S$$ in $$\{1,2,\ldots,n\}$$ such that $$S\cap A=\emptyset$$ and $$\lvert S\rvert >\varepsilon n$$?

The best solution was submitted by Huy Tùng Nguyễn (2016학번). Congratulations!

Here is his solution of problem 2017-8.

Alternative solutions were submitted by 조태혁 (수리과학과 2014학번, +3), 위성군 (수리과학과 2015학번, +3), 최인혁 (물리학과 2015학번, +3, solution), 오동우 (수리과학과 2015학번, +3), 최대범 (수리과학과 2016학번, +3), 이본우 (2017학번, +3), 김태균 (수리과학과 2016학번, +3), Ivan Adrian Koswara (전산학부 2013학번, +3), 이재우 (함양고등학교 2학년, +3), 장기정 (수리과학과 2014학번, +2).

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# Solution: 2017-05 Inequality for a continuous function

Suppose that $$f : (2, \infty) \to (-2, 2)$$ is a continuous function and there exists a positive constant $$m$$ such that $$| 1 + xf(x) + (f(x))^2 | \leq m$$ for any $$x > 2$$. Prove that, for any $$x > 2$$,
$\left| f(x) – \frac{\sqrt{x^2 -4}-x}{2} \right| \leq 6 \sqrt{m}.$

The best solution was submitted by Huy Tùng Nguyễn (2016학번). Congratulations!

Here is his solution of problem 2017-05.

Alternative solutions were submitted by 위성군 (수리과학과 2015학번, +3), 조태혁 (수리과학과 2014학번, +3), 최인혁 (물리학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), 오동우 (수리과학과 2015학번, +3), 이본우 (2017학번, +3), 김재현 (수리과학과 2016학번, +3), 김태균 (수리과학과 2016학번, +2).

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