2008 Indonesia MO Problems/Problem 3
Solution 1
Summing up the equation yields the result
. Thus,
Since ,
,
, are pairwise relatively prime, this implies that
,
, and
.
, but because
and
,
. Similarly,
and
.
WLOG, .
Suppose .
Since they are pairwise relatively prime, and all ,
,
,
is not equal to
nor
, and
is not equal to
. A strict order of
can be made. Since
and
, we have
. We also know that
and
, which implies that
. Thus, the only way
, while
, is if
.
Plugging in , we get
, and
. Because
, we know that
. Thus, in order for
and
, the only possible way is for
. However, we have previously established that
, and combined with the fact that
,
and
are not co-prime, which is a contradiction.
Hence, .
Case 1:
Let . since
, we have
. The only options will be
and
. This gives us the answers
and
Case 2:
Let . If
, then
, and they won't be co-prime. As a result,
is strictly less than
. Since
, and
, we have
. Similarly,
. But
. Thus,
. Using the fact that
, we have
. Hence,
, and
.
Plugging in the answers ,
, and
all yield valid results of
,
, and
, respectively.