PUMaC 2014 · 数论（A 组） · 第 5 题

PUMaC 2014 — Number Theory (Division A) — Problem 5

[ 5 ] Find the number of pairs of integer solution ( x, y ) that satisfies the equation ( x − y + 2)( x − y − 2) = − ( x − 2)( y − 2)

解析

[ 5 ] Find the number of pairs of integer solution ( x, y ) that satisfies the equation ( x − y + 2)( x − y − 2) = − ( x − 2)( y − 2) Solution: 1 2 2 2 Reorganize and we get ( x − y ) + ( x − 2) + ( y − 2) = 8. The only possible scenarios are when two of the three terms on the left evaluate to 4, and the other one to zero. Every scenario gives two solutions: when x − y = 0, x = y = 4 or x = y = 0; when x − 2 = 0, x = 2 , y = 4 or x = 2 , y = 0; when y − 2 = 0, x = 4 , y = 2 or x = 0 , y = 2.Hence there are 6 pairs of solution.