Find all real solutions x, y, z of the following system of equations:

x³ + y = 12x + 20
2y³ + z = 24y + 36
3z³ + x = 36z + 52

Seeing the regularity in the equations, and the difference (16) between 20 and 36, and 36 and 52, led me to think about powers of 2, and quickly got x=y=z=4 as an answer.

Then, I decided to write y as a function of x as per the first equation; substitute in the second, and write z as a function of x, and finally substitute in the third, that became an enormous polynomial in x... which I didn't want to solve!

Posted by Federico Kereki on 2006-07-30 16:27:10