A Great Math Olympiad Problem that Looks Complicated / Germany 

x=32/y , --> , y^2 - 12y + 32=0 , --> , y^2-8y -4y+32=0 , (y-8)(y-4)=0 , y= 8 , 4 , x=12-y , case 1 , y=8 , x=12-8 , x=4 , +1 -8 case 2 , y=4 , x=12-4 , x=8 , -4 +32=0 , solu , (x , y) , (4 , 8) , (8 , 4) , test , x*y=32 , 4*8=8*4 , --> , 32 , OK ,

xy=32=1×32=2×16=4×8 Here 4+8=12 (x,y)=(4,8) or (8,4)

x = 8 y = 4 8 + 4 = 12 8 x 4 = 32