An algebra problem by Efren Medallo

Algebra Level 5

Let f ( x ) = x 3 \mathrm{f}(x) = \sqrt[3]{x} defined over R \mathbb{R} .

If f ( f ( 2 ) − 1 ) = f ( a ) − f ( b ) + f ( c ) \mathrm{f}( \mathrm{f}(2) - 1) = \mathrm{f}(a) - \mathrm{f}(b) + \mathrm{f}(c) for positive rational numbers a > b > c a>b>c , and that a + b + c = P Q a+b+c = \frac{ P}{Q} for coprime positive integers P P and Q Q , determine P + Q P+Q .


The answer is 16.

Efren Medallo by Efren Medallo , 26, Philippines

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1 solution

Efren Medallo
Jun 4, 2017

Remember, m 3 ± n 3 = ( m ± n ) ( m 2 ∓ m n + n 2 ) \color{#3D99F6} m^3 \pm n^3 = (m \pm n)(m^2 \mp mn + n^2) \color{#333333} .

S = f ( f ( 2 ) − 1 ) = 2 3 − 1 3 S = \mathrm{f}(\mathrm{f}(2) -1) = \sqrt[3]{\sqrt[3]{2} - 1}

S = ( 2 3 − 1 ) ⋅ 2 3 2 + 2 3 + 1 2 3 2 + 2 3 + 1 3 S = \sqrt[3]{ (\sqrt[3]{2} - 1) \cdot \frac{ \sqrt[3]{2}^2 + \sqrt[3]{2} + 1}{\sqrt[3]{2}^2 + \sqrt[3]{2} + 1}}

S = 2 − 1 2 3 2 + 2 3 + 1 3 S = \sqrt[3]{\frac{2-1}{\sqrt[3]{2}^2 + \sqrt[3]{2} + 1}}

S = 1 2 3 2 + 2 3 + 1 3 S = \sqrt[3]{\frac{1}{\sqrt[3]{2}^2 + \sqrt[3]{2} + 1}}

S = 3 3 ( 2 3 2 + 2 3 + 1 ) 3 S = \sqrt[3]{\frac{3}{3(\sqrt[3]{2}^2 + \sqrt[3]{2} + 1)}}

S = 3 ( 2 + 1 ) ( 2 3 2 + 2 3 + 1 ) 3 S = \sqrt[3]{\frac{3}{(2+1)(\sqrt[3]{2}^2 + \sqrt[3]{2} + 1)}}

S = 3 ( 2 3 + 1 ) ( 2 3 2 − 2 3 + 1 ) ( 2 3 2 + 2 3 + 1 ) 3 S = \sqrt[3]{\frac{3}{(\sqrt[3]{2}+1)(\sqrt[3]{2}^2 - \sqrt[3]{2} + 1)(\sqrt[3]{2}^2 + \sqrt[3]{2} + 1)}}

S = 3 ( 2 3 + 1 ) ( 2 3 4 + 2 3 2 + 1 ) 3 S = \sqrt[3]{\frac{3}{(\sqrt[3]{2}+1)(\sqrt[3]{2}^4 + \sqrt[3]{2}^2 + 1)}}

S = 3 ( 2 3 + 1 ) ( 2 2 3 + 2 3 2 + 1 ) 3 S = \sqrt[3]{\frac{3}{(\sqrt[3]{2}+1)(2 \sqrt[3]{2} + \sqrt[3]{2}^2 + 1)}}

S = 3 ( 2 3 + 1 ) ( 2 3 2 + 2 2 3 + 1 ) 3 S = \sqrt[3]{\frac{3}{(\sqrt[3]{2}+1)( \sqrt[3]{2}^2 + 2 \sqrt[3]{2} + 1)}}

S = 3 ( 2 3 + 1 ) ( 2 3 + 1 ) 2 3 S = \sqrt[3]{\frac{3}{(\sqrt[3]{2}+1)(\sqrt[3]{2} + 1)^2 }}

S = 3 ( 2 3 + 1 ) 3 3 S = \sqrt[3]{\frac{3}{(\sqrt[3]{2}+1)^3}}

S = 3 3 2 3 + 1 S = \frac{ \sqrt[3]{3} }{ \sqrt[3]{2} + 1}

S = 3 3 3 2 ( 2 3 + 1 ) S = \frac{ 3}{ \sqrt[3]{3}^2 ( \sqrt[3]{2} + 1) }

S = 2 + 1 9 3 ( 2 3 + 1 ) S = \frac{2+1}{ \sqrt[3]{9} ( \sqrt[3]{2} + 1) }

S = ( 2 3 + 1 ) ( 2 3 2 − 2 3 + 1 ) 9 3 ( 2 3 + 1 ) S = \frac{(\sqrt[3]{2} + 1)(\sqrt[3]{2}^2 -\sqrt[3]{2} + 1)}{ \sqrt[3]{9} ( \sqrt[3]{2} + 1) }

S = 2 3 2 − 2 3 + 1 9 3 S = \frac{\sqrt[3]{2}^2 - \sqrt[3]{2} + 1}{\sqrt[3]{9}}

S = 4 9 3 − 2 9 3 + 1 9 3 S = \sqrt[3]{ \frac{4}{9}} - \sqrt[3]{ \frac{2}{9}} + \sqrt[3]{ \frac{1}{9}}

S = f ( a ) − f ( b ) + f ( c ) S = \mathrm{f}(a) - \mathrm{f}(b) + \mathrm{f}(c)

a + b + c = 7 9 a + b + c = \frac{7}{9}

P + Q = 16 P+Q= 16

5 Helpful 0 Interesting 0 Brilliant 0 Confused

Such an epic solution!

Kelvin Hong - 4 years ago

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Thanks! :)

Efren Medallo - 4 years ago

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