7. Find all pairs of natural numbers ( $x$, y) for which the equation $x^{2}+x y=y+92$ holds.

OTBET: $(2,88) ;(8,4)$.
Solution: Transform $x^{2}-1+x y-y=91$. Factorize: $(x-1)(x+y+1)=7 \times 13$. Both factors are positive, and the first factor must be smaller, so the possible cases are $x-1=1, x+y+1=91$ or $x-1=7$, $x+y+1=13$. We get the solutions $x=2, y=88$ or $x=8, y=4$.

Lomonosov Moscow State University

## School Olympiad "Conquer Sparrow Hills" in Mathematics

Final Round Tasks for the 2015/2016 Academic Year for 9th Grade