Factorization is an art(7)

Algebra Level 5

{ x 3 − 3 x y 2 − 119 = 0 y 3 − 3 x 2 y − 120 = 0 x 2 + y 2 = z 4 3 \large \begin{cases} x^3-3xy^2-119 &=0 \\ y^3-3x^2y-120 &=0 \\ x^2+y^2 &= z^{\frac{4}{3}}\end{cases}

If z z is an positive integer, find z z which satisfy the system of equations above.


The answer is 13.

Tommy Li by Tommy Li , 22, Hong Kong

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3 solutions

敬全 钟
Jun 15, 2017

Since ( x 3 − 3 x 2 y ) 2 + ( y 3 − 3 x 2 y ) 2 = ( x 2 + y 2 ) 3 (x^3-3x^2y)^2+(y^3-3x^2y)^2=(x^2+y^2)^3 and z z is a positive integer, we see that z 4 = ( x 2 + y 2 ) 3 = 11 9 2 + 12 0 2 ⇒ z = 28561 4 = 13. z^4=(x^2+y^2)^3=119^2+120^2\Rightarrow z=\sqrt[4]{28561}=13. @Tommy Li, nice problem!

6 Helpful 0 Interesting 0 Brilliant 0 Confused

Simply put x=z^(2/3)sinA

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Nacer Jaafar
Jun 15, 2017

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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