MathDB

A5

Part of 1963 Putnam

Problems(1)

Putnam 1963 A5

Source: Putnam 1963

5/1/2022
i) Prove that if a function ff is continuous on the closed interval [0,π][0, \pi] and ∫0πf(t)cos⁡t  dt=∫0πf(t)sin⁡t  dt=0, \int_{0}^{\pi} f(t) \cos t \; dt= \int_{0}^{\pi} f(t) \sin t \; dt=0, then there exist points 0<α<β<π0 < \alpha < \beta < \pi such that f(α)=f(β)=0.f(\alpha) =f(\beta) =0.
ii) Let RR be a bounded, convex, and open region in the Euclidean plane. Prove with the help of i) that the centroid of RR bisects at least three different chords of the boundary of R. R.
Putnamfunctiontrigonometry