perplexus.info :: Just Math : Only One, One, One II

Only One, One, One II (Posted on 2013-12-08)

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.

See The Solution Submitted by K Sengupta

Rating: 2.3333 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

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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