# Solution: 2019-09 Discrete entropy

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).

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# Solution: 2018-22 Two monic quadratic polynomials

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.

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