IMO 1969 LL CZS14

Let a and b be two positive real numbers. If x is a real solution

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IMO 1969 LL CZS14

Origin: CZS

Problem

Let a and b be two positive real numbers. If x is a real solution of the equation x2 + px + q = 0 with real coefficients p and q such that |p| \leqa, |q| \leqb, prove that |x| \leq1  a + a2 + 4b . (1) Conversely, if x satisfies (1), prove that there exist real numbers p and q with |p| \leqa, |q| \leqb such that x is one of the roots of the equation x2 + px + q = 0.