Wait, did my post not post? Grrr.

Let x <= y <= z.

x+y+z<=3z, thus xy<=3, thus x<=sqrt(3), thus x=1.

1+1+z=z doesn't work, so y=2 or y=3.

Yada yada, something about the Jackson 5, and there's your answer!

(Engineering method: Had MATLAB probe [1,100]^3, found 6 answers. Bada bing, bada boom!)