2019 IMO Problems/Problem 4
Problem
Find all pairs of positive integers such that
Solution 1
(when ), (when ), (when )
(when ), (when )
Hence, , satisfy
For is strictly increasing, and will never satisfy = 2 for integer n since when .
In all solutions, for any prime and positive integer , we will denote by the exponent of the largest power of that divides . The right-hand side of will be denoted by that is,
=
On the other hand, is expressed by the as
Thus, implies the inequality In order to obtain an opposite estimate, observe that We claim that for all
For the estimate (3) is true because and ~flamewavelight and ~phoenixfire
See Also
2019 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |