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Omni-MATH / Let X_(r)=x^(r)+y^(r)+z^(r) with x,y,z real. It is known that if S₁=0, (*)quadfrac{S_{m+n}}{m+n}=frac{S_m}{m}frac{S_n}{n} for (m,n)=(2,3),(3,2),(2,5), or (5,2). Determine [i]all[/i…

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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, \[(*)\quad\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}\] for (m,n)=(2,3),(3,2),(2,5)(m,n)=(2,3),(3,2),(2,5), or (5,2)(5,2). Determine [i]all[/i] 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.
Plain-text mathematical notation (without MathML)
Let X_(r)=x^(r)+y^(r)+z^(r) with x,y,z real. It is known that if S₁=0, \[(*)\quad\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}\] for (m,n)=(2,3),(3,2),(2,5), or (5,2). Determine [i]all[/i] other pairs of integers (m,n) if any, so that $(*)$ holds for all real numbers x,y,z such that x+y+z=0.
Original LaTeX notation
Let $X_r=x^r+y^r+z^r$ with $x,y,z$ real. It is known that if $S_1=0$, \[(*)\quad\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}\] for $(m,n)=(2,3),(3,2),(2,5)$, or $(5,2)$. Determine [i]all[/i] other pairs of integers $(m,n)$ if any, so that $(*)$ holds for all real numbers $x,y,z$ such that $x+y+z=0$.

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task_id
67c934b7-0ae4-573d-ad6b-e1c987202e01
task_key
test--67c934b7-0ae4-573d-ad6b-e1c987202e01
task_revision_id
1
config / split
/ test

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