IMO Shortlist 1969 problem 14


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2. travnja 2012.
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(CZS 3) Let a and b be two positive real numbers. If x is a real solution of the equation x^2 + px + q = 0 with real coefficients p and q such that |p| \le a, |q| \le b, prove that |x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b}) Conversely, if x satisfies the above inequality, prove that there exist real numbers p and
q with |p|\le a, |q|\le b such that x is one of the roots of the equation x^2+px+ q = 0.
Izvor: Međunarodna matematička olimpijada, shortlist 1969