# Solution June 02, 2007

### Problem

Find all polynomials of degree 3, such that for each

### Solution

Let

- .

For all we have

First let *x* = *y* = 0 to see that .

Now set *x* = *y* and let *x* go to infinity we see from **(1)** that .

Consider the inequality

**(2)** follows from **(1)** for all by letting . On the other hand, if we let
then **(1)** follows from **(2)** and the arithmetic-geometric mean
inequality, .

If , we find that by letting *t* go to infinity, and this suffices to satisfy **(2)**.

Assume that . We want to find a condition on *d* such that **(2)** is always satisfied. We start by minimizing *q*(*t*) for , so we set its derivative to 0:

Note that so only minima greater than 0 need to be considered.

If , then , so there are no other minima to consider.

If , then

is negative in this case, so this is a stronger condition on *d* than we already had.

In summary, *p*(*x*) is a polynomial satisfying for all in the following cases:

- , and
- , and

No other possibilities for *p*(*x*) exist.