Library / Ukrainian National Mathematical Olympiad, 3rd Round /14 of 24

Ukrainian National Mathematical Olympiad, 3rd Round

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.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

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