Monthly Archives: March 2018

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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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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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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