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Omni-MATH / Find all integer n such that the following property holds: for any positive real numbers a,b,c,x,y,z, with max(a,b,c,x,y,z)=a , a+b+c=x+y+z and abc=xyz, the inequality …

Problem

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problem

Find all integer nn such that the following property holds: for any positive real numbers a,b,c,x,y,za,b,c,x,y,z, with max(a,b,c,x,y,z)=amax(a,b,c,x,y,z)=a , a+b+c=x+y+za+b+c=x+y+z and abc=xyzabc=xyz, the inequality an+bn+cn≥xn+yn+zna^n+b^n+c^n \ge x^n+y^n+z^n holds.
Plain-text mathematical notation (without MathML)
Find all integer n such that the following property holds: for any positive real numbers a,b,c,x,y,z, with max(a,b,c,x,y,z)=a , a+b+c=x+y+z and abc=xyz, the inequality a^(n)+b^(n)+c^(n)≥x^(n)+y^(n)+z^(n) holds.
Original LaTeX notation
Find all integer $n$ such that the following property holds: for any positive real numbers $a,b,c,x,y,z$, with $max(a,b,c,x,y,z)=a$ , $a+b+c=x+y+z$ and $abc=xyz$, the inequality $$a^n+b^n+c^n \ge x^n+y^n+z^n$$ holds.

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Source and history

task_id
a1eee54b-61a9-57b9-9257-761f94633330
task_key
test--a1eee54b-61a9-57b9-9257-761f94633330
task_revision_id
4
config / split
/ test

Official source

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