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1982 USAMO #2

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August 16, 2011
AMCUSA(J)MOUSAMOnumber theory unsolvednumber theory

Problem Statement

Let Xr=xr+yr+zrX_r=x^r+y^r+z^r with x,y,zx,y,z real. It is known that if S1=0S_1=0, (*) \frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n} for (m,n)=(2,3),(3,2),(2,5)(m,n)=(2,3),(3,2),(2,5), or (5,2)(5,2). Determine all other pairs of integers (m,n)(m,n) if any, so that (∗)(*) holds for all real numbers x,y,zx,y,z such that x+y+z=0x+y+z=0.
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