7. Find all pairs of integers ( $x, y$ ) for which the equation $x^{2}+y^{2}=x+y+2$ holds.

ANSWER: $(-1,0) ;(-1,1) ;(0,-1) ;(0,2) ;(1,-1),(1,2),(2,0) ;(2,1)$
Solution: The function $2-\mathrm{x}$ takes values from -0.25 to infinity, $-\mathrm{y} 2+\mathrm{y}+2$ from minus infinity to 2.25. Common values (integers) are $0,1,2$, further by trial.

Lomonosov Moscow State University

## School Students' Olympiad "Conquer Sparrow Hills" in Mathematics

Final stage tasks for the 2015/2016 academic year for 9th grade