Putnam 1963 A5

i) Prove that if a function $f$ is continuous on the closed interval $[0, \pi]$ and $$\int_{0}^{\pi} f(t) \cos t dt= \int_{0}^{\pi} f(t) \sin t dt=0,$$ then there exist points $0 < \alpha < \beta < \pi$ such that $f(\alpha) =f(\beta) =0.$ ii) Let $R$ be a bounded, convex, and open region in the Euclidean plane. Prove with the help of i) that the centroid of $R$ bisects at least three different chords of the boundary of $R.$