Determine all possible pairs (x,y) of positive integers such that:
x+1 y+1
Each of ---- and ----- is a positive integer.
y x
**** Adapted from a problem appearing in Polish Mathematical Olympiad.
Combining the fractions, we get
(xy + x + y + 1) /xy must be an integer
Subtracting 1, we get that
(x + y + 1) /xy must be an integer
If x = 1, then
(y + 2)/y must be an integer, so 2/y must be an integer,
so y can only be 1 or 2.
This leads to solutions (1,1), (1,2) and by symmetry (2,1)
If x = 2, then
(y + 3)/2y must be an integer
y can only be 1 or 3,
This leads to additional solution (2,3) and by symmetry (3,2)
And there are no other solutions, because if x and y are both greater than 3, then (x + y + 1) /xy is less than 1
Analytical solution
