2021 JMPSC Invitationals Problems/Problem 5
Problem
An -pointed fork is a figure that consists of two parts: a handle that weighs
ounces and
"skewers" that each weigh a nonzero integer weight (in ounces). Suppose
is a positive integer such that there exists a fork with weight
What is the sum of all possible values of
?
Solution
If each skewer weights ounces, where
must be a positive integer, then the total weight of our fork is
We equate this to
and rearrange to get
If
is an integer and
is not, it is clear that
will not be an integer. Thus, since
is an integer, the only possible values of
that yield an integer
are factors of
:
Note that
is negative for
and so the only valid
are
leading to an answer of
. ~samrocksnature