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Omni-MATH / Let S_(r)=x^(r)+y^(r)+z^(r) with x,y,z real. It is known that if S₁=0 ,

Problem

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problem

Let

S_r=x^r+y^r+z^r

with

x,y,z

real. It is known that if

S_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)

, or

(5,2)

. Determine all other pairs of integers

(m,n)

if any, so that $(*)$ holds for all real numbers

x,y,z

such that

x+y+z=0

Plain-text mathematical notation (without MathML)

Let S_(r)=x^(r)+y^(r)+z^(r) with x,y,z real. It is known that if S₁=0 ,
$(*)$  (S_(m+n))/(m+n)=(S_(m))/(m)(S_(n))/(n) 
for (m,n)=(2,3),(3,2),(2,5) , or (5,2) . Determine all 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 $S_r=x^r+y^r+z^r$ with $x,y,z$ real. It is known that if $S_1=0$ ,
$(*)$  $\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}$ 
for $(m,n)=(2,3),(3,2),(2,5)$ , or $(5,2)$ . Determine all 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: 25bf8255-7cd3-5acf-8611-cdc5a25a1bac
task_key: test--25bf8255-7cd3-5acf-8611-cdc5a25a1bac
task_revision_id: 2
config / split: / test

Official source

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