OIMU 2006 Problem 4

Prove that for any interval $[a,b]$ of real numbers and any positive integer $n$ there exists a positive integer $k$ and a partition of the given interval $$a = x (0) < x (1) < x (2) < \cdots < x (k-1) < x (k) = b$$ such that $$\int_{x(0)}^{x(1)}f(x)dx+\int_{x(2)}^{x(3)}f(x)dx+\cdots=\int_{x(1)}^{x(2)}f(x)dx+\int_{x(3)}^{x(4)}f(x)dx+\cdots$$ for all polynomials $f$ with real coefficients and degree less than $n$ .