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MMO 285 Moscow MO 1954 x^2+Ax+B=0, x^2+Cx+D=0, x^2+(A+C)/2 x+(B+D)/2=0

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August 13, 2019
algebraquadraticstrinomialabsolute valueroots

Problem Statement

The absolute values of all roots of the quadratic equation x2+Ax+B=0x^2+Ax+B = 0 and x2+Cx+D=0x^2+Cx+D = 0 are less then 11. Prove that so are absolute values of the roots of the quadratic equation x2+A+C2x+B+D2=0x^2 + \frac{A + C}{2} x + \frac{B + D}{2} = 0.
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