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Only One, One, One II (Posted on 2013-12-08) Difficulty: 3 of 5
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)

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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 = (A2B2 + B2C2 + C2A2)/3

Now using the inequality: arithmetic mean >= geometric mean 

=>   ABC >= ((A2B2)(B2C2)(C2A2))^(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 2013-12-13 14:04:04
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