Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.8 AIME Prove it Soviet Union

Problem:

Find three non-zero reals such that all quadratics with those numbers as coefficients have two distinct rational roots.

Solution

Solution:

Answer $1$ , $2$ , $-3$

If $a + b + c = 0$ , then $1$ is a root of $a x^2 + b x + c$ , and so the other root is $-\frac{b}{a} - 1$ , which is rational.

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