Competition problem — Algebra — Maths Olympiad Prep

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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.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.