# 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.

• 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? – String Mar 26 '15 at 14:05
• 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. – hOff Mar 26 '15 at 14:48
• Thank you for elaborating on that! – String Mar 26 '15 at 14:53