Difference between revisions of "2013 Mock AIME I Problems/Problem 7"
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== Solution == | == Solution == | ||
− | <math>\boxed{005}</math>. | + | |
+ | Note that the only non-primitive <math>7</math>th or <math>9</math>th root of unity with a positive imaginary part is <math>e^{i\tfrac{6\pi}9} = e^{i\tfrac{2\pi}3}</math>. Listing the other such roots shows that both <math>S</math> and <math>T</math> have <math>3</math> elements, so <math>C</math> is equal to <math>3</math> times the sum of all the elements of <math>S</math> plus <math>3</math> times the sum of all the elements of <math>T</math>, because all of the terms are added thrice to the sum. | ||
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+ | Because all of the <math>7</math>th roots of unity sum to <math>0</math>, the sum of their real parts must be <math>0</math>. Without <math>1</math>, the sum of their real parts is <math>-1</math>. Because reciprocals of <math>7</math>th roots of unity are also <math>7</math>th roots of unity (but with opposite imaginary parts and the same real part), the sum of the real parts of the roots with a positive imaginary part must be <math>-\tfrac1 2</math>. The same ''would'' be true for the primitive <math>9</math>th roots of unity, but we have to remember to exclude <math>e^{i\tfrac{2\pi}3}</math>, which, by [[Euler's Identity]], has a real part of <math>\cos(\tfrac{2\pi}3)=-\tfrac 1 2</math>. Subtracting this from <math>-\tfrac 1 2</math> yields <math>0</math>, so the sum of the real parts of the elements of <math>T</math> is <math>0</math>. | ||
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+ | Thus, <math>|\Re(C)| = |3(-\tfrac 1 2)| = \tfrac 3 2</math>, so our answer is <math>3+2=\boxed{005}</math>. | ||
== See also == | == See also == |
Latest revision as of 13:56, 30 July 2024
Problem
Let be the set of all
th primitive roots of unity with imaginary part greater than
. Let
be the set of all
th primitive roots of unity with imaginary part greater than
. (A primitive
th root of unity is a
th root of unity that is not a
th root of unity for any
.)Let
. The absolute value of the real part of
can be expressed in the form
where
and
are relatively prime numbers. Find
.
Solution
Note that the only non-primitive th or
th root of unity with a positive imaginary part is
. Listing the other such roots shows that both
and
have
elements, so
is equal to
times the sum of all the elements of
plus
times the sum of all the elements of
, because all of the terms are added thrice to the sum.
Because all of the th roots of unity sum to
, the sum of their real parts must be
. Without
, the sum of their real parts is
. Because reciprocals of
th roots of unity are also
th roots of unity (but with opposite imaginary parts and the same real part), the sum of the real parts of the roots with a positive imaginary part must be
. The same would be true for the primitive
th roots of unity, but we have to remember to exclude
, which, by Euler's Identity, has a real part of
. Subtracting this from
yields
, so the sum of the real parts of the elements of
is
.
Thus, , so our answer is
.