2006 USAMO Problems/Problem 4
Problem
Find all positive integers such that there are
positive rational numbers
satisfying
.
Solution
First, consider composite numbers. We can then factor into
It is easy to see that
, and thus, we can add
1s in order to achieve a sum and product of
. For
, which is only possible in one case,
, we consider
.
Secondly, let be a prime. Then we can find the following procedure: Let
and let the rest of the
be 1. The only numbers we now need to check are those such that
. Thus, we need to check for
. One is included because it is neither prime nor composite.
For , consider
. Then by AM-GM,
for
. Thus,
is impossible.
If , once again consider
. Similar to the above,
for
since
and
. Obviously,
is then impossible.
If , let
. Again,
. This is obvious for
. Now consider
. Then
is obviously greater than
. Thus,
is impossible.
If , proceed as above and consider
. Then
and
. However, we then come to the quadratic
, which is not rational. For
and
we note that
and
. This is trivial to prove. If
, it is obviously impossible, and thus
does not work.
The last case, where , is possible using the following three numbers.
shows that
is possible.
Hence, can be any positive integer greater than
with the exclusion of
.
See Also
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