BMO1 2009/10 Problem 6 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.

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.