Curt Monash

The 2009 Putnam Exam may be found at http://amc.maa.org/a-activities/a7-problems/putnam... A presumably reliable answer sheet may be found at http://amc.maa.org/a-activities/a7-problems/putnam... But the answer I come up with differs.

The question is:

Say that a polynomial with real coefficients in two variables, x, y, is balanced if the average value of the polynomial on each circle centered at the origin is 0. The balanced polynomials of degree at most 2009 form a vector space V over R. Find the dimension of V.

My answer is as follows:

Divide the monomials x^j * y^k of degree <= 2009 into 4 disjoint sets:

S1: j, k both are even.

S2: j is even, k is odd

S3: j is odd, k is even

S4: j, k are both odd

The union of S1, S2, S3, and S4 is obviously a basis for the entire vector space of polynomials in x,y with degree <= 2009.

Now, the members of S1 are all positive at any point that is not the origin, while the members of S2, S3, and S4 all average to 0 on any circle centered at the origin. So the union of S2, S3, and S4 is a basis for V, and dimension of V is just the sum of their cardinalities -- i.e., 3 * 1005^2.

But the answer sheet says the answer is 2 * 1005^2. Where did I go wrong?

Let f(n) be a sequence of polynomials defined simply as f(n)(x)=x^n.

The f(m)(f(n)) (x) = f(mn)(x) for all m,n, and hence f(m)(f(n)) = f(n)(f(m)). So far, so obvious.

So what about OTHER sequences of polynomials with those properties? I've found one other sequence, also with integer coefficients, of polynomials in one variable with the properties:

A. f(n) is a polynomial of degree n.

B. f(n)(f(m)) = f (mn)

C. f(n)(f(m)) = f(m)(f(n))

So my questions are:

1. Prove, as I have, that there exists at least one other sequence of polynomials with properties A, B, and C beyond the trivial case given above.

2. Are there any other sequences of polynomials with those properties?

3. Are there any with properties A and C but not B?

4. Do the answers change if we allow coefficients to be, say, from the field of real (or, if you prefer, complex) numbers?

There really is a best answer to this, at least in a mathematics section. :D

I'm struggling with another problem from Rotman, namely:

If a group G has exactly two conjugacy classes, and at least one element of finite order, prove that it is isomorphic to the cyclic group of order 2.

Here's what I've got.

A. Let a be an element of order n. Then any other non-identity element is a conjugate of a, and hence has order dividing n.

B. By a very similar argument, all non-identity elements have the same order.

C. Obviously, that order must be a prime.

D. If that prime equals 2, the result is easy to prove as follows:

For any non-identity a and b, ab has order 2 (or 1). So abab = 1. So bab = a.

We just proved that all conjugates of a equal a. Since every non-identity element of G is by assumption a conjugate of a, we're done.

---------------------------------------------------------------

So what am I missing?

Let S and T be (not necessarily disjoint) subsets of a finite group G. Prove that either G = ST or |G| >= |S| + |T|.

I just can't get a grip on where to start with that problem.

How many ways can 81,000 be written as the sum of two positive integer squares? Prove your answer.

:)

Hint: Some of my recent prior questions.

List the unique solutions to 1000 = a^2 + b^2, where a and b are positive integers. Prove that that's all of them.

Hint: Some of the questions I've asked recently.

Prove your answer.

True or false:

An integer prime p can be written as the sum of two squares if and only if there exists an integer n such that n^2 is congruent to -1 (modulo p).

Prove your answer in a fairly elementary way.

In the finite field Zp -- i.e., the integers modulo p, for p a prime -- "-1" is always defined. For which p does -1 have a square root?

State your test and give an elementary proof.

Which integer primes are also primes in the Gaussian Integers?

I'll be quite impressed if somebody who hasn't seen this question before works out not just the correct answer, but a solid proof for it as well.

We define an interesting number to be a number that has at least one interesting property.

Prove that all natural numbers are interesting.

What novel started "Not to every young girl is it given to enter the harem of the Sultan of Turkey and return to her homeland a virgin"?

Hint: The homeland was England, near the Scottish border.