Difference between revisions of "2021 JMPSC Invitationals Problems/Problem 5"
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+ | == Solution 3 == | ||
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+ | Suppose the weight of each "skewer" is <math>k</math> ounces. We then have that the total weight of the fork is <math>12 + nk</math>. This must equal <math>n^2</math>, so <math>12 + nk = n^2 \implies 12 = n^2 - nk = 12 \implies n(n-k) = 12</math>. This means that <math>n</math> must be a factor of <math>12</math>. Also, the total weight must be greater than <math>12</math> ounces, so we have that <math>n^2 > 12</math>. The factors of <math>12</math> that have a square greater than <math>12</math> are <math>4</math>, <math>6</math> and <math>12</math>, so the answer is <math>4+6+12=\boxed{22}</math>. | ||
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==See also== | ==See also== |
Latest revision as of 13:00, 12 July 2021
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 an -pointed 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
Solution 2
Suppose the integer weight is : we have . Now, we have , so we can have , , and to ensure is positive. Therefore,
~Geometry285
Solution 3
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 .
~Mathdreams
See also
- Other 2021 JMPSC Invitationals Problems
- 2021 JMPSC Invitationals Answer Key
- All JMPSC Problems and Solutions
The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition.