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Nine quadratics

Source: All-Russian 2011

May 17, 2011
quadraticsalgebra proposedalgebraQuadratic

Problem Statement

Nine quadratics, x2+a1x+b1,x2+a2x+b2,...,x2+a9x+b9x^2+a_1x+b_1, x^2+a_2x+b_2,...,x^2+a_9x+b_9 are written on the board. The sequences a1,a2,...,a9a_1, a_2,...,a_9 and b1,b2,...,b9b_1, b_2,...,b_9 are arithmetic. The sum of all nine quadratics has at least one real root. What is the the greatest possible number of original quadratics that can have no real roots?
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