## Putnam 2012 Day 1

1. Let be real numbers in the open interval Show that there exist distinct indices such that are the side lengths of an acute triangle.

2. Let be a commutative and associative binary operation on a set Assume that for every and in there exists in such that (This may depend on and ) Show that if are in and then

3. Let be a continuous function such that

(i) for every in

(ii) and

(iii) exists and is finite.

Prove that is unique, and express in closed form.

4. Let and be integers with and let and be intervals on the real line. Let be the set of all where and are integers with in and let be the set of all integers in such that is in Show that if the product of the lengths of and is less than then is the intersection of with some arithmetic progression.

5. Let denote the field of integers modulo a prime and let be a positive integer. Let be a fixed vector in let be an matrix with entries in and define by Let denote the -fold composition of with itself, that is, and Determine all pairs for which there exist and such that the vectors are distinct.

6. Let be a continuous, real-valued function on Suppose that, for every rectangular region of area the double integral of over equals Must be identically ?

**Hints: **1. Note that (I’m talking here about the Fibonacci sequence).

2. You can prove that the law has an identity element, and then it is almost finished.