a a , b b , and c c

Number Theory Level 2

∣ 1 1 + a + a b + 1 1 + b + b c + 1 1 + c + a c ∣ \large \left| \frac { 1 }{ 1+a+ab } +\frac { 1 }{ 1+b+bc } +\frac { 1 }{ 1+c+ac } \right|

Real numbers a a , b b , and c c are such that a b c = 1 abc = 1 , determine the value of the expression above.


The answer is 1.

Wildan Bagus Wicaksono by Wildan Bagus Wicaksono , 18, Indonesia

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1 solution

Chew-Seong Cheong
Jul 16, 2017

x = ∣ 1 1 + a + a b + 1 1 + b + b c + 1 1 + c + c a ∣ = ∣ 1 1 + a + a b + a a + a b + a b c + a b a b + a b c + a b c a ∣ = ∣ 1 1 + a + a b + a a + a b + 1 + a b a b + 1 + a ∣ = ∣ 1 ∣ = 1 \begin{aligned} x & = \left|\frac 1{1+a+ab} + \frac 1{1+b+bc} + \frac 1{1+c+ca}\right| \\ & = \left|\frac 1{1+a+ab} + \frac a{a+ab+{\color{#3D99F6}abc}} + \frac {ab}{ab+{\color{#3D99F6}abc} +{\color{#3D99F6}abc}a}\right| \\ & = \left|\frac 1{1+a+ab} + \frac a{a+ab+{\color{#3D99F6}1}} + \frac {ab}{ab+{\color{#3D99F6}1} +a}\right| \\ & = |1| \\ & = \boxed{1} \end{aligned}

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Your solution is easier hehe.

Wildan Bagus Wicaksono - 3 years, 10 months ago

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There is usually a trick. We have to look for it.

Chew-Seong Cheong - 3 years, 10 months ago

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