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split in black and white intervals for quadratic polynomials

Source: China TST 1995, problem 6

May 17, 2005
quadraticsalgebrapolynomialalgebra unsolved

Problem Statement

Prove that the interval [0,1]\lbrack 0,1 \rbrack can be split into black and white intervals for any quadratic polynomial P(x)P(x), such that the sum of weights of the black intervals is equal to the sum of weights of the white intervals. (Define the weight of the interval [a,b]\lbrack a,b \rbrack as P(b)āˆ’P(a)P(b) - P(a).) Does the same result hold with a degree 3 or degree 5 polynomial?
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