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Only One, One, One II (
Posted on 20131208
)
A=B=C=1 is a
positive real
solution of:
A*B B*C C*A  +  +  = 3 C A B
Prove that it is the only such solution.
No Solution Yet
Submitted by
K Sengupta
Rating:
1.0000
(2 votes)
Comments: (
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)
Assuming positive integers (spoiler)
Comment 5 of 5 
Assuming A,B and C are positive integers....
3 = AB/C + BC/A + CA/B
=>
ABC = (A
^{2}
B
^{2}
+ B
^{2}
C
^{2}
+ C
^{2}
A
^{2}
)/3
Now using the inequality: arithmetic mean >= geometric mean
=>
ABC >= ((A
^{2}
B
^{2}
)(B
^{2}
C
^{2}
)(C
^{2}
A
^{2}
))^(1/3)
=>
ABC >= (ABC)^(4/3)
=>
1 >= (ABC)^(1/3)
=>
1 >= ABC
So A=B=C=1 is possible, but none of A,B,C can be greater than 1.
Posted by
Harry
on 20131213 14:04:04
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