AoPS Wiki talk:Problem of the Day/July 26, 2011

Solution:

Move all of the terms to one side:

$x^3 + 369xy + y^3 + 123^3 = 0$

Now, factor:

$\frac{(x + y + 123)((x - y)^2 + (y - 123)^2 + (-123 + x)^2)}{2} = 0$

Multiply by $2$ and get rid of the negative in front of $123$ :

$(x + y + 123)((x - y)^2 + (y - 123)^2 + (123 - x)^2) = 0$

One possibility is that $x + y = -123$ , yielding us an infinite number of possibilities, giving us an answer of $\infty$ .

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