THE ART AND SCIENCE OF MATHEMATICS Wednesday, April 19, 2006

Three Questions

1. Evaluate
   
   $$\int_{-\infty}^{+\infty} \frac{\cos x}{1+x^2} dx$$
   
   

2. Assume an object goes a distance one unit in time one, and that it starts and finishes at rest. Show that it must have either accelerated or decelerated at a rate of at least four units at some point.

3. You are given twelve coins, eleven of which are identical and one which is either heavier or lighter. Given nothing but a balance, determine which coin is bad, and whether it is heavier or lighter. You are allowed at most three comparisons.