An algebra problem by Rishik Jain

Algebra Level 2

If a b c = 1 abc=1 , find

1 1 + a + b 1 + 1 1 + b + c 1 + 1 1 + c + a 1 \dfrac{1}{1 + a + b^{-1}} + \dfrac{1}{1 + b + c^{-1}} + \dfrac{1}{1 + c + a^{-1}}


The answer is 1.

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1 solution

Sajjad Sajjad
Mar 23, 2015

By simplifying the given form we get,

b b + a b + 1 + c c + b c + 1 a a + c a + 1 \frac{b}{b + ab + 1} + \frac{c}{c + bc +1} \frac{a}{a + ca + 1}

As, a b c = 1 abc = 1

a b = 1 c ab = \frac{1}{c} and a = 1 b c a = \frac{1}{bc}

Puting the value we get,

b c b c + c + 1 + c b c + c + 1 + 1 b c + c + 1 \frac{bc}{bc + c + 1} + \frac{c}{bc + c + 1} + \frac{1}{bc + c + 1}

= b c + c + 1 b c + c + 1 = \frac{bc + c + 1}{bc + c + 1}

= 1 = 1 :)

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