2021 Fall AMC 12B Problems/Problem 16
Problem
Suppose ,
,
are positive integers such that
and
What is the sum of all possible distinct values of
?
Solution
Let ,
,
. WLOG, let
. We can split this off into cases:
: let
we can try all possibilities of
and
to find that
is the only solution.
: No solutions. By
and
, we know that
,
, and
have to all be divisible by
. Therefore,
cannot be equal to
.
: C has to be both a multiple of
and
. Therefore,
has to be a multiple of
. The only solution for this is
.
: No solutions. By
and
, we know that
,
, and
have to all be divisible by
. Therefore,
cannot be equal to
.
: No solutions. By
and
, we know that
,
, and
have to all be divisible by
. Therefore,
cannot be equal to
.
: No solutions. By
and
, we know that
,
, and
have to all be divisible by
. Therefore,
cannot be equal to
.
: No solutions. As
,
, and
have to all be divisible by
,
has to be divisible by
. This contradicts the sum
.
Putting these solutions together, we have
-ConcaveTriangle