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Putnam 1939 B6

Source:

August 20, 2021
Putnam

Problem Statement

Do either (1)(1) or (2)(2):
(1)(1) ff is continuous on the closed interval [a,b][a, b] and twice differentiable on the open interval (a,b).(a, b). Given x0∈(a,b),x_0 \in (a, b), prove that we can find ξ∈(a,b)\xi \in (a, b) such that ((f(x0)−f(a))(x0−a)−(f(b)−f(a))(b−a))(x0−b)=f′′(ξ)2.\dfrac{ ( \dfrac{(f(x_0) - f(a))}{(x_0 - a)} - \dfrac{(f(b) - f(a))}{(b - a)} )}{(x_0 - b)} = \dfrac{f''(\xi)}{2}.
(2)(2) ABAB and CDCD are identical uniform rods, each with mass mm and length 2a.2a. They are placed a distance bb apart, so that ABCDABCD is a rectangle. Calculate the gravitational attraction between them. What is the limiting value as a tends to zero?
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