Track / Stage 4 / 314 of 340 #574 of 1964

Problem 574

AMC 12 late, AIME early

Number theory Difficulty 5.0 Find the answer

$\left.\begin{array}{l}{\left[\begin{array}{l}\text { equations in integers }\end{array}\right]} \\ \text { [Completing the square. Sums of squares }]\end{array}\right]$

Solve the equation $x^{2}+y^{2}=x+y+2$ in integers.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Official solution

Let's write the equation as $(2 x-1)^{2}+(2 y-1)^{2}=10$ .

From this, it is clear that one term equals 9, and the other equals 1.

## Answer

$\{2,0\},\{2,1\},\{-1,0\},\{-1,1\}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.