Competition problem — Algebra — Olympiad Maths Prep

Library / /17 of 29

Algebra Difficulty 5.2 AIME, harder Prove it Ukraine

Find all values of parameter $b$ , such that for all $x$ at least one function $f_1(x) = x^2 + 2011x + b$ or $f_2(x) = x^2 - 2011x + b$ is positive.

Solution

For $x = 0$ we have $f_1(0) = f_2(0) = b$ , thus all $b \le 0$ does not satisfy the condition of the problem.

Let $b > 0$ . We add two values up and get $f_1(x) + f_2(x) = 2x^2 + 2b > 0$ , thus at least one function is positive.

Looking for a route rather than an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.