Suppose f(x) is a quadratic polynomial with leading coefficient 1 such that f(x) + f(-x) = 2x^2 for all x and f(2016) = 0. What is the value of f(10) ?

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  1. if f(x) = x^2+bx+c, we have

    (x^2+bx+c) + (x^2-bx+c) = 2x^2
    2x^2+c = 2x^2
    so, c=0 and

    f(x) = x^2+bx
    f(2016) = 0, so
    2016^2 + 2016b = 0
    b = -2016

    f(x) = x^2 - 2016x = x(2016-x)

    f(10) = 10(2016-10) = 10(2006) = 20060

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