Putnam 1972 A6

Let $f$ be an integrable real-valued function on the closed interval $[0, 1]$ such that $$\int_{0}^{1} x^{m}f(x) dx=\begin{cases} 0 \; \text{for} m=0,1,\ldots,n-1; 1\; \text{for} m=n. \end{cases}$$ Show that $|f(x)|\geq2^{n}(n+1)$ on a set of positive measure.