Putnam Problem Set 2: Fall '99

1. Evaluate .

2. Let be any set of 20 distinct integers chosen from the arithmetic progression 1,4, 7, ..., 100. Prove that there must be two distinct integers in whose sum is 104.

3. In a round-robin tournament with players , , ..., (where >1), each player plays one game with each of the other players and the rules are such that no ties can occur. Let and be the number of games won and lost, respectively, by . Show that

.

4. Show that, for any sequence , , ... of real numbers, the two conditions

and

are equivalent.

5. If and are continuous and periodic functions with period 1 on the real line, then show that

.

6. Define to be 1. For , let be the number of by matrices whose elements are nonnegative integers with the property that , ( = 1, 2, ..., ), and where

= 1 ( = 1,2, .., ). Prove

(a) = +

(b) .