2018-23 Game of polynomials

Two players play a game with a polynomial with undetermined coefficients
\[
1 + c_1 x + c_2 x^2 + \dots + c_7 x^7 + x^8.
\]
Players, in turn, assign a real number to an undetermined coefficient until all coefficients are determined. The first player wins if the polynomial has no real zeros, and the second player wins if the polynomial has at least one real zero. Find who has the winning strategy.

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.

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 \( f_2(x)\le 0\).

Solution: 2018-21 AM-GM inequality

Does there exist a (possibly \(n\)-dependent) constant \( C \) such that
\[
\frac{C}{a_n} \sum_{1 \leq i < j \leq n} (a_i-a_j)^2 \leq \frac{a_1+ \dots + a_n}{n} – \sqrt[n]{a_1 \dots a_n} \leq \frac{C}{a_1} \sum_{1 \leq i < j \leq n} (a_i-a_j)^2
\]
for any \( 0 < a_1 \leq a_2 \leq \dots \leq a_n \)?

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

Here is his solution of problem 2018-21.

Alternative solutions were submitted by 하석민 (수리과학과 2017학번, +3),
이본우 (수리과학과 2017학번, +2). One incorrect submission was received.

2018-21 AM-GM inequality

Does there exist a (possibly \(n\)-dependent) constant \( C \) such that
\[
\frac{C}{a_n} \sum_{1 \leq i < j \leq n} (a_i-a_j)^2 \leq \frac{a_1+ \dots + a_n}{n} - \sqrt[n]{a_1 \dots a_n} \leq \frac{C}{a_1} \sum_{1 \leq i < j \leq n} (a_i-a_j)^2 \] for any \( 0 < a_1 \leq a_2 \leq \dots \leq a_n \)?

Solution: 2018-20 Almost Linear Function

Let \(f:\mathbb R\to\mathbb R\) be a function such that \[ -1\le f(x+y)-f(x)-f(y)\le 1\] for all reals \(x\), \(y\). Does there exist a constant \(c\) such that \( \lvert f(x)-cx\rvert \le 1\) for all reals \(x\)?

The best solution was submitted by Ha, Seokmin (하석민, 수리과학과 2017학번). Congratulations!

Here is his solution of problem 2018-20.

An alternative solution was submitted by 채지석 (수리과학과 2016학번, +3). There were two incorrect submissions.

Solution: 2018-19 Gauss’s theorem

Let
\[
f(x) = 1 + \left( \frac{1}{2} \cdot x \right)^2 + \left( \frac{1}{2} \cdot \frac{3}{4} \cdot x^2 \right)^2 + \left( \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot x^3 \right)^2 + \dots
\]
Prove that
\[
(\sin x) f(\sin x) f'(\cos x) + (\cos x) f(\cos x) f'(\sin x) = \frac{2}{\pi \sin x \cos x}.
\]

The best solution was submitted by Seo, Juneyoung (서준영, 수리과학과 대학원생). Congratulations!

Here is his solution of problem 2018-19.

Alternative solutions were submitted by 길현준 (2018학번, +3, solution), 김기현 (수리과학과 대학원생, +3), 이본우 (수리과학과 2017학번, +3).

2018-19 Gauss’s theorem

Let
\[
f(x) = 1 + \left( \frac{1}{2} \cdot x \right)^2 + \left( \frac{1}{2} \cdot \frac{3}{4} \cdot x^2 \right)^2 + \left( \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot x^3 \right)^2 + \dots
\]
Prove that
\[
(\sin x) f(\sin x) f'(\cos x) + (\cos x) f(\cos x) f'(\sin x) = \frac{2}{\pi \sin x \cos x}.
\]