# Math Help - Problem 34

1. ## Problem 34

Given $x,y,z\in \mathbb{Z}$ solve the Diophantine equation:
$x^3+y^3+z^3 = (x+y+z)^3$.

2. All permutations of $(k, -k, j)$ for integers k and j are solutions. To see that these are the only solutions, rewrite as $y^3+z^3 = (x+y+z)^3 - x^3$ which, assuming $y \ne -z$, reduces to $x^2+(y+z)x+yz = 0 \Rightarrow x = -y$ or $x = -z$ with the other variable arbitrary, giving the solutions $(k, -k, j), \ (k, j, -k)$ and, by symmetry in the original equation, $(j, k, -k)$

3. Is it possible to propose the next 'problem of the week'?

4. Apparently I can't post new threads, so I'll just append a problem here.

Source: The Art of Problem Solving (Vol 2)

Show that for any two positive numbers, RMS-AM >= GM-HM where RMS denotes the root-mean square, AM the arithmetic mean, GM the geometric mean, and HM the harmonic mean.

5. Originally Posted by albi
Is it possible to propose the next 'problem of the week'?
I try to get one by Sunday or Monday.

Apparently I can't post new threads, so I'll just append a problem here.
Blocked priveleges.

6. Originally Posted by ThePerfectHacker
I try to get one by Sunday or Monday.
Um. I think we didnt understand ourselves.

I meant I have one idea which may become ´problem of the week´.

7. Originally Posted by albi
Um. I think we didnt understand ourselves.

I meant I have one idea which may become ´problem of the week´.
Of course you can! We're always open to suggestions. The Problem of the Week writers (CB and TPH) work very hard though so be constructive!