# 2018-05 Roulette

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.

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# Solution: 2018-03 Integers from square roots

Find all integers $$n$$ such that $$\sqrt{1} + \sqrt{2} + \dots + \sqrt{n}$$ is an integer.

The best solution was submitted by Han, Junho (한준호, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2018-03.

Alternative solutions were submitted by 김태균 (수리과학과 2016학번, +3), 이본우 (수리과학과 2017학번, +3), 이종원 (수리과학과 2014학번, +3, solution), 채지석 (수리과학과 2016학번, +3), 최백규 (2016학번, +3), 최인혁 (물리학과 2015학번, +3), 김건우 (수리과학과 2017학번, +2). Two incorrect solutions were received.

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# 2018-04 An inequality

Let $$x_1,x_2,\ldots,x_n$$ be reals such that $$x_1+x_2+\cdots+x_n=n$$ and $$x_1^2+x_2^2+\cdots +x_n^2=n+1$$. What is the maximum of $$x_1x_2+x_2x_3+x_3x_4+\cdots + x_{n-1}x_n+x_nx_1$$?

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# Solution: 2018-02 Impossible to squeeze

For $$n\ge 1$$, let $$f(x)=x^n+\sum_{k=0}^{n-1} a_k x^k$$ be a polynomial with real coefficients. Prove that if $$f(x)>0$$ for all $$x\in [-2,2]$$, then $$f(x)\ge 4$$ for some $$x\in [-2,2]$$.

The best solution was submitted by Choi, Inhyeok (최인혁, 물리학과 2015학번). Congratulations!

Here is his solution of problem 2018-02.

Alternative solutions were submitted by 이본우 (수리과학과 2017학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 이종원 (수리과학과 2014학번, +3), 채지석 (수리과학과 2016학번, +3), 하석민 (수리과학과 2017학번, +3), 한준호 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +2), 이재우 (함양고등학교 3학년, +2).

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# 2018-03 Integers from square roots

Find all integers $$n$$ such that $$\sqrt{1} + \sqrt{2} + \dots + \sqrt{n}$$ is an integer.

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# Solution: 2018-01 Recurrence relation

Define a sequence $$\{ a_n \}$$ by $$a_1 = a$$ and
$a_n = \frac{2n-1}{n-1} a_{n-1} -1$
for $$n \geq 2$$. Find all real values of $$a$$ such that $$\lim_{n \to \infty} a_n$$ exists.

The best solution was submitted by Bonwoo Lee (이본우, 수리과학과 2017학번). Congratulations!

Here is his solution of problem 2018-01.

Alternative solutions were submitted by 강한필 (전산학부 2016학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 이종원 (수리과학과 2014학번, +3), 채지석 (수리과학과 2016학번, +3), 최인혁 (물리학과 2015학번, +3), 한준호 (수리과학과 2015학번, +3), 고성훈 (2018학번, +2), 김태균 (수리과학과 2016학번, +2), 송교범 (고려대 수학과 2017학번, +2), 이재우 (함양고등학교 3학년, +2), 노우진 (물리학과 2015학번) 및 윤정인 (물리학과 2016학번) (+2). Two incorrect solutions were received.

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# 2018-02 Impossible to squeeze

For $$n\ge 1$$, let $$f(x)=x^n+\sum_{k=0}^{n-1} a_k x^k$$ be a polynomial with real coefficients. Prove that if $$f(x)>0$$ for all $$x\in [-2,2]$$, then $$f(x)\ge 4$$ for some $$x\in [-2,2]$$.

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Define a sequence $$\{ a_n \}$$ by $$a_1 = a$$ and
$a_n = \frac{2n-1}{n-1} a_{n-1} -1$
for $$n \geq 2$$. Find all real values of $$a$$ such that $$\lim_{n \to \infty} a_n$$ exists.