MathDB
Putnam 1939 B5

Source:

August 20, 2021
Putnam

Problem Statement

Do either (1)(1) or (2)(2):
(1)(1) Prove that ∫1k[x]f′(x)dx=[k]f(k)−∑1[k]f(n),\int_{1}^{k} [x] f'(x) dx = [k] f(k) - \sum_{1}{[k]} f(n), where k>1,k > 1, and [z][z] denotes the greatest integer ≤z.\leq z. Find a similar expression for: ∫1k[x2]f′(x)dx.\int_{1}^{k} [x^2] f'(x) dx.
(2)(2) A particle moves freely in a straight line except for a resistive force proportional to its speed. Its speed falls from 1,000fts1,000 \dfrac{ft}{s} to 900fts900 \dfrac{ft}{s} over 1200ft.1200 ft. Find the time taken to the nearest 0.01s.0.01 s. [No calculators or log tables allowed!]
Back to ProblemsView on AoPS