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Putnam 1963 A5

Source: Putnam 1963

May 1, 2022
Putnamfunctiontrigonometry

Problem Statement

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.
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