2021 JMPSC Invitationals Problems/Problem 5
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 an -pointed fork with weight What is the sum of all possible values of ?
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
Suppose the integer weight is : we have . Now, we have , so we can have , , and to ensure is positive. Therefore,
Suppose the weight of each "skewer" is ounces. We then have that the total weight of the fork is . This must equal , so . This means that must be a factor of . Also, the total weight must be greater than ounces, so we have that . The factors of that have a square greater than are , and , so the answer is .
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