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Long John Silverman has captured a treasure map from Adam McBones. Adam has buried the treasure at the point $(x,y)$ with integer co-ordinates (not necessarily positive). He has indicated on the map the values of $x^2 + y$ and $x + y^2$, and these numbers are distinct. Prove that Long John has to dig only in one place to find the treasure.

Thanks in advance for any contributions.

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Lets we have two places to dig: $$x_1^2 + y_1 = x_2^2 + y_2 \space (1)$$ and $$ x_1 + y^2_1 = x_2 + y^2_2 \space (2)$$ Rewrite equations as: $$\frac{x_1 - x_2}{y_2 - y_1} = \frac{1}{x_1 + x_2} $$ and $$\frac{x_1 - x_2}{y_2 - y_1} = y_2 + y_1$$ Because $y_2 + y_1$ is integer we conclude that $|x_1+x_2| = 1$ ; $|y_1 + y_2| = 1$

Then consider separate cases when $x_1+x_2 = 1$ and $x_1+x_2 = -1$ Plugging each case into equation (1) and (2) we will see that $$x_1^2 + y_1 = x_1 + y^2_1$$ that is contradiction.

Interesting point here is that we cannot find numbers, only proof that they are unique.

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  • $\begingroup$ I am not sure I understand why your last equation would follow by plugging in each case to (1) and (2). Would you mind helping me understand that part? $\endgroup$ – String Mar 26 '15 at 14:05
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    $\begingroup$ Let $x_1 = x, y_1 = y$, then (first case) $x_2 = 1 - x, y_2 = 1 - y$. Solving (1), (2) we will get that $y_1 = 1 - x$ and Adam's number are the same. $\endgroup$ – hOff Mar 26 '15 at 14:48
  • $\begingroup$ Thank you for elaborating on that! $\endgroup$ – String Mar 26 '15 at 14:53

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