α is a real number. Find all continuous real-valued functions f :[0, 1] → (0, ∞) such that ∫01 f(x) dx = 1, ∫01 x f(x) dx = α, ∫01 x2 f(x) dx = α2.
We have ∫01 (α - x)2f(x) dx = α2 - 2α2 + α2 = 0. But the integrand is positive, except possibly for one point of the range, so the integral must also be positive. Contradiction. So there are no functions with this property.
25th Putnam 1964
© John Scholes
5 Feb 2002