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2021 JMPSC Invitationals Problems/Problem 5

Problem

An $n$-pointed fork is a figure that consists of two parts: a handle that weighs $12$ ounces and $n$ "skewers" that each weigh a nonzero integer weight (in ounces). Suppose $n$ is a positive integer such that there exists a fork with weight $n^2.$ What is the sum of all possible values of $n$?

Solution

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