Putnam 1939 B5

Take either (i) or (ii). (i) Prove that $\int_{1}^{a}[x] f^{\prime}(x) d x=[a] f(a)-\{f(1)+\cdots+f([a])\}$ where $a$ is greater than 1 and where [ $x$ ] denotes the greatest of the integers not exceeding $x$ . Obtain a corresponding expression for $\int_{1}^{a}\left[x^{2}\right] f^{\prime}(x) d x$ (ii) A particle moves on a straight line, the only force acting on it being a resistance proportional to the velocity. If it started with a velocity of $1,000 \mathrm{ft}$ . per sec. and had a velocity of 900 ft . per sec. when it had travelled $1,200 \mathrm{ft}$ ., calculate to the nearest hundredth of a second the time it took to travel this distance.

Dataset text PutnamGAP. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.