Consider the cards with labels \( 1,\dots, n \) in some order. If the top card has label \(m \), we reverse the order of the top \( m \) cards. The process stops only when the card with label \( 1\) is on the top. Prove that the process must stop in at most \( (1.7)^n \) steps.

The best solution was submitted by 길현준 (수리과학과 2018학번). Congratulations!

Here is his solution of problem 2020-18.

Other solutions was submitted by 김유일 (2020학번, +3), 이준호 (수리과학과 2016학번, +3).

Suppose that \( X \) is a discrete random variable on the set \( \{ a_1, a_2, \dots \} \) with \( P(X=a_i) = p_i \). Define the discrete entropy
\[
H(X) = -\sum_{n=1}^{\infty} p_i \log p_i.
\]
Find constants \( C_1, C_2 \geq 0 \) such that
\[
e^{2H(X)} \leq C_1 Var(X) + C_2
\]
holds for any \( X \).

The best solution was submitted by 길현준 (2018학번). Congratulations!

Here is his solution of problem 2019-09.

Alternative solutions were submitted by 최백규 (생명과학과 2016학번, +3). Incomplete solutions were submitted by, 이정환 (수리과학과 2015학번, +2), 채지석 (수리과학과 2016학번, +2).

Let \(f_1(x)=x^2+a_1x+b_1\) and \(f_2(x)=x^2+a_2x+b_2\) be polynomials with real coefficients. Prove or disprove that the following are equivalent.

(i) There exist two positive reals \(c_1, c_2\) such that \[ c_1f_1(x)+ c_2 f_2(x) > 0\] for all reals \(x\).

(ii) There  is no real \(x\) such that \( f_1(x)\le 0\) and \( f_2(x)\le 0\).

The best solution was submitted by Gil, Hyunjun (길현준, 2018학번). Congratulations!

Here is his solution of problem 2018-22.

Alternative solutions were submitted by 김태균 (수리과학과 2016학번, +3), 서준영 (수리과학과 대학원생, +3), 이본우 (수리과학과 2017학번, +3), 채지석 (수리과학과 2016학번, +3), 하석민 (수리과학과 2017학번, +3), 최백규 (생명과학과 2016학번, +2). There was one incorrect submission.