$a,b$ are real numbers that satisfy
$$a^3 - 3ab^2 = 29$$
$$b^3 - 3a^2b = 34$$
What is the value of $a^2 + b^2$?
Meta puzzle: Which year did this problem appear?
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Solution:
The solution becomes simple if you think complex!
$$(a+ib)^3 = (a^3 - 3ab^2) - i(b^3 - 3a^2b)$$
Thus $$(a+ib)^3 = 29 - 34i$$
And so $$(a^2 + b^2)^3 = 29^2 + 34^2 = 1997$$
and
$$a^2 + b^2 = \sqrt[3]{1997}$$
Note that we basically have the identity
$$(a^2 + b^2)^3 = (a^3 - 3ab^2)^2 + (b^3 - 3a^2b)^2$$
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