Difference between revisions of "2021 JMPSC Invitationals Problems/Problem 5"
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If each skewer weights <math>a</math> ounces, where <math>a</math> must be a positive integer, then the total weight of our fork is <math>12+an.</math> We equate this to <math>n^2</math> and rearrange to get <cmath>12+an=n^2</cmath> <cmath>an=n^2-12</cmath> <cmath>a=n-\frac{12}{n}.</cmath> If <math>n</math> is an integer and <math>\frac{12}{n}</math> is not, it is clear that <math>a</math> will not be an integer. Thus, since <math>n</math> is an integer, the only possible values of <math>n</math> that yield an integer <math>a</math> are factors of <math>12</math>: <cmath>n=1,2,3,4,6,12.</cmath> Note that <math>a</math> is negative for <math>n=1,2,3</math> and so the only valid <math>n</math> are <math>4,6,12,</math> leading to an answer of <math>4+6+12=\boxed{22}</math>. ~samrocksnature | If each skewer weights <math>a</math> ounces, where <math>a</math> must be a positive integer, then the total weight of our fork is <math>12+an.</math> We equate this to <math>n^2</math> and rearrange to get <cmath>12+an=n^2</cmath> <cmath>an=n^2-12</cmath> <cmath>a=n-\frac{12}{n}.</cmath> If <math>n</math> is an integer and <math>\frac{12}{n}</math> is not, it is clear that <math>a</math> will not be an integer. Thus, since <math>n</math> is an integer, the only possible values of <math>n</math> that yield an integer <math>a</math> are factors of <math>12</math>: <cmath>n=1,2,3,4,6,12.</cmath> Note that <math>a</math> is negative for <math>n=1,2,3</math> and so the only valid <math>n</math> are <math>4,6,12,</math> leading to an answer of <math>4+6+12=\boxed{22}</math>. ~samrocksnature | ||
+ | ==Solution 2== | ||
+ | Suppose the integer weight is <math>k</math>: we have <math>n^2-nk-12=0</math>. Now, we have <math>12=2^2 \cdot 3</math>, so we can have <math>(n-12)(n+1)</math>, <math>(n-6)(n+2)</math>, and <math>(n-4)(n+3)</math> to ensure <math>k</math> is positive. Therefore, <math>n=\{4,6,12 \} \implies 4+6+12=\boxed{22}</math> | ||
+ | ~Geometry285 | ||
==See also== | ==See also== |
Revision as of 21:09, 11 July 2021
Contents
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
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.