Putnam 2012 Day 2
1. Let be a class of functions from to that satisfies:
(i) The functions and are in
(ii) If and are in the functions and are in
(iii) If and are in and for all then the function is in
Prove that if and are in then the function is also in .
2. Let be a given (non-degenerate) polyhedron. Prove that there is a constant with the following property: If a collection of balls whose volumes sum to contains the entire surface of then .
3. A round-robin tournament among teams lasted for days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the games. Over the course of the tournament, each team played every other team exactly once. Can one necessarily choose one winning team from each day without choosing any team more than once?
4. Suppose that and that for Does have a finite limit as (Here )
5. Prove that, for any two bounded functions there exist functions such that for every
6. Let be an odd prime number such that Define a permutation of the residue classes modulo by Show that is an even permutation if and only if
Hints: 1. To prove that if we choose two function in our set their product is still there is just a matter of cleverly passing from the functions to their logarithms, then to sum the logarithms obtained, thus obtaining the logarithm of a product, take the compositioin with the exponential to get rid of the logarithm and use the last property about the subtraction.
2. Use only one face, and the fact that the intersection of a sphere of radius with a plane surface has area at most the area of a great circle of the sphere. Then use a well chosen inequality to finish.
4. Denote and apply Stolz Cesaro.