2005 IMO Shortlist Problems/N3
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Problem
(Mongolia)
Let ,
,
,
,
, and
be positive integers. Suppose that the sum
divides both
and
. Prove that
is composite.
This was also Problem 1 of the 2nd 2006 German TST, and a problem at the 2006 Indian IMO Training Camp.
Solution
For all integers we have
,
since each coefficient of the first two polynomials is congruent to the corresponding coefficient of the second two polynomials, mod . Now, suppose
is prime. Since
,
one of is divisible by
, say
. Since
, this means
. But since
are positive integers, we then have
,
a contradiction. ∎
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.