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Harvard-MIT Mathematics Tournament

Algebra Difficulty 4.9 AIME Prove it United States

Problem:

Let $a$ , $b$ , and $c$ be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: $a x^{2} + b x + c$ , $b x^{2} + c x + a$ , and $c x^{2} + a x + b$ .

Solution

Solution:

Answer: 4

If all the polynomials had real roots, their discriminants would all be nonnegative: $a^{2} \geq 4 b c$ , $b^{2} \geq 4 c a$ , and $c^{2} \geq 4 a b$ . Multiplying these inequalities gives $(a b c)^{2} \geq 64(a b c)^{2}$ , a contradiction. Hence one of the quadratics has no real roots.

The maximum of 4 real roots is attainable: for example, the values $(a, b, c) = (1, 5, 6)$ give $-2, -3$ as roots to $x^{2} + 5 x + 6$ and $-1, -\frac{1}{5}$ as roots to $5 x^{2} + 6 x + 1$ .

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