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.