2001 Pan African MO Problems/Problem 1
Problem
Find all positive integers such that:
is a positive integer.
Solution
Perform polynomial long division to get . Note that if
, then
can not be an integer. Thus, all of the solutions satisfy the inequality
.
If , then
. However, there are no positive integers in this case. If
, then
and
. Rearranging the second inequality results in
. Factoring results in
, so
.
Now there are only four possible positive integers, so we can use trial and error to determine if is a positive integer. After doing trial and error, the only positive integers that make
an integer are
or
.
See Also
2001 Pan African MO (Problems) | ||
Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All Pan African MO Problems and Solutions |