Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.0 AIME Prove it Bulgaria

The functions $f(x) = 2x^2 + 2x - 4$ and $g(x) = x^2 - x + 2$ are given. Find all real values of $x$ such that:

a) $\frac{f(x)}{g(x)}$ is a positive integer;

b) the inequality $\sqrt{f(x)} + \sqrt{g(x)} \ge \sqrt{2}$ holds.

Solution

a) Hint. Set $\frac{f(x)}{g(x)} = k$ , where $k$ is a positive integer. Then $(2-k)x^2 + (2+k)x - 2(2+k) = 0$ and use the fact that the discriminant of this quadratic equation is nonnegative.

Answer. $x = \frac{-3+\sqrt{33}}{2}, \frac{-3-\sqrt{33}}{2}, 2$ .

b) Answer. $x \in (-\infty, -2] \cup [1, +\infty)$ .

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