# Solution: 2018-15 Diophantine equation

Let $$n$$ be a positive integer. Suppose that $$a_1, a_2, \dots, a_n$$ are non-zero integers and $$b_1, b_2, \dots, b_n$$ are positive integers such that $$(b_i, b_n) = 1$$ for $$i = 1, 2, \dots, n-1$$. Prove that the Diophantine equation
$a_1 x_1^{b_1} + a_2 x_2^{b_2} + \dots + a_n x_n^{b_n} = 0$
has infinitely many integer solutions $$(x_1, x_2, \dots, x_n)$$.

The best solution was submitted by Chae, Jiseok (채지석, 수리과학과 2016학번). Congratulations!

Here is his solution of problem 2018-15.

An alternative solution was submitted by Saba Dzmanashvili (수리과학과 2017학번, +3), 강한필 (전산학부 2016학번, +3), 권홍 (중앙대 물리학과, +3), 이본우 (수리과학과 2017학번, +3), 이재우 (함양고등학교 3학년, +3), 최백규 (생명과학과 2016학번, +3), 길현준 (2018학번, +2), 김태균 (수리과학과 2016학번, +2).

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# Solution: 2018-13 Bernoulli vectors

Assume that $$x \in \mathbb{R}^n$$ with at least $$k$$ non-zero entries $$( k> 0 )$$. Let
$A = \{ y \in \{-1, 1\}^n : y \cdot x = 0 \}.$
Prove that $$|A| \leq k^{-1/2} 2^n$$.

The best solution was submitted by Chae, Jiseok (채지석, 수리과학과 2016학번). Congratulations!

Here is his solution of problem 2018-13.

An alternative solution was submitted by 이대석 (수리과학과 2017학번, +3). Two incorrect solutions were received.

GD Star Rating # Concluding 2018 Spring

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

1st prize (Gold): Lee, Jongwon (이종원, 수리과학과 2014학번)
2nd prize (Silver): Chae, Jiseok (채지석, 수리과학 과 2016학번)
2nd prize (Silver): Han, Joon Ho (한준호,수리과학과 2015학번)
2nd prize (Silver): Lee, Bonwoo (이본우, 수리과학과 2017학번)
3rd prize (Bronze): Ko, Sunghun (고성훈, 2018학번)

이종원 (수리과학과 2014학번) 40/40
채지석 (수리과학과 2016학번) 35/40
한준호 (수리과학과 2015학번) 35/40
이본우 (수리과학과 2017학번) 32/40
고성훈 (2018학번) 20/40
김태균 (수리과학과 2016학번) 19/40
최인혁 (물리학과 2015학번) 10/40
김건우 (수리과학과 2017학번) 8/40
최백규 (생명과학과 2016학번) 6/40
하석민 (수리과학과 2017학번) 6/40
길현준 (2018학번) 3/40
강한필 (전산학부 2016학번) 3/40
문정욱 (2018학번) 3/40
노우진 (물리학과 2015학번) 1/40
윤정인 (물리학과 2016학번) 1/40

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# Solution: 2018-10 Probability of making an acute triangle

Given a stick of length 1, we choose two points at random and break it into three pieces. Compute the probability that these three pieces form an acute triangle.

The best solution was submitted by Chae, Jiseok (채지석, 수리과학과 2016학번). Congratulations!

Here is his solution of problem 2018-10.

Alternative solutions were submitted by 고성훈 (2018학번, +3), 이종원 (수리과학과 2014학번, +3), 한준호 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +2), 이본우 (수리과학과 2017학번, +2), 이준성 (상문고등학교 2학년, +2), Harrison Zhu (Imperial College London, +2).

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A gambler is playing roulette and betting $1 on black each time. The probability of winning$1 is 18/38, and the probability of losing $1 is 20/38. Find the probability that starting with$20 the player reaches \$40 before losing the money.