2013 JBMO Problems/Problem 1
Problem
Find all ordered pairs of positive integers for which the numbers and are both positive integers
Solution
Adding to both the given numbers we get:
is also a positive integer so we have:
= is a positive integer
Similarly,
is also a positive integer so we have:
= is a positive integer
Combining above results we get:
which is a valid solution.
which are valid solutions.
Thus, all solutions are:
Solution 1.5 (credit to dskull16)
To get the two results:
We can also add zero to the numerator as follows:
since
since
Then proceed as above.