You can't Set a = b = c a=b=c

Algebra Level 4

Minimize a 3 ( a − b ) ( a − c ) + b 3 ( b − c ) ( b − a ) + c 3 ( c − a ) ( c − b ) \large \frac{a^3}{(a-b)(a-c)} + \frac{b^3}{(b-c)(b-a)}+\frac{c^3}{(c-a)(c-b)}

Given that a , b , c ∈ R + a,b,c \in \mathbb{R^+} and a b c = 1 abc=1 .

Note : Give your answer to two decimal places.


The answer is 3.00.

Vilakshan Gupta by Vilakshan Gupta , 19, India

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1 solution

Alan Yan
Jan 27, 2018

The expression simplifies to a + b + c a+b+c which from AM-GM is at least 3 3 . We can get arbitrary close to 3 3 by letting a , b , c a, b, c tend to each other.

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