Difference between revisions of "AoPS Wiki talk:Problem of the Day/August 2, 2011"
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<cmath>(x_2,y_2) = (\sqrt[3]{218 + \psi^3},\psi)</cmath> | <cmath>(x_2,y_2) = (\sqrt[3]{218 + \psi^3},\psi)</cmath> | ||
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| + | In <math>\mathbb{C}^2</math>, there are more solutions. | ||
Revision as of 04:03, 2 August 2011
Problem: Find all 
such that 
and 
.
Solution: Let 
and 
. Then 
and 
. Hence,
![\[218+y^3=(24+y^2)^3\]](http://latex.artofproblemsolving.com/9/7/e/97ef276c7c9c24cdd1e14fef5a80a661dc215cbe.png)
![\[(y-5)(6740+1348y-76y^2+72y^3)=0\]](http://latex.artofproblemsolving.com/2/9/6/296e700e4e33655622769940263ec443b1ac3d2e.png)
This quartic has two real roots; 
and
![\[y = \frac{19}{54} - \frac{17837}{54(-7881974+25353\sqrt{105481})^{1/3}}+\frac{1}{54}(-7881974+25353\sqrt{105481})^{1/3}\]](http://latex.artofproblemsolving.com/8/e/4/8e47388f34bf739d28bff629c84d96e403430c45.png)
by the cubic formula. Call this second root (which evaluates to about -3.01) 
. The two real solutions are therefore:
![\[(x_1,y_1) = (7,5)\]](http://latex.artofproblemsolving.com/6/e/b/6eba84427df0ea51b99491e9488db4f120d81e22.png)
and
![\[(x_2,y_2) = (\sqrt[3]{218 + \psi^3},\psi)\]](http://latex.artofproblemsolving.com/9/e/4/9e41e4e53819e6165753a60fd58f85889fa1f0e9.png)
In 
, there are more solutions.
