2020 CIME I Problems/Problem 6: Difference between revisions

Latest revision as of 11:18, 1 February 2025

Problem 6

Find the number of complex numbers $z$ satisfying $|z|=1$ and $z^{850}+z^{350}+1=0$ .

Solution 1

We reduce the problem to $z^{17}+z^7+1$ , remembering to multiply the final product by 50. We need the imaginary parts of the numbers $z^{17},z^7$ to cancel, which by working modulo 360 we can easily determine only happens when the number is of the form $cis(15x)$ (this holds true because we are only looking for solutions with a magnitude of $1$ ). We also need the real parts to sum to $-1$ . We check all the multiples of 15 that result in $cis(x)$ being negative, and find that only two work(or alternatively, if you are good, you can guess that only $120$ and $240$ work). The answer is then $100$ .

Note

One definite way to find $cisa$ and $cisb$ such that they sum to $-1$ is by setting up equations.

Let cos $a=x$ and cos $b=y$ . By definition

$x+y=-1$ $\sqrt{1-x^2}+\sqrt{1-y^2}=0$

We can ingnore plus or minus for the sins as this would still yield the same result. From the second equation $1-x^2=1-y^2$ and $x= \pm y$ .

Plugging into the first equation, we see that $x=-1/2, y=-1/2$ . And we have $a=120$ and $b=240$ or the other way around.

(Note by Bigbrain_2009)

Solution 2

Let $f(w)=w^{850}+w^{350}+1$ and suppose $z$ is such that $f(z)=0$ and $|z|=1$ . Note that $z$ is not real (by descartes rule of signs), thus $\overline{z}=|z|^2/z=1/z$ is also a root of $f$ . It follows that $z^{850}f(1/z)=z^{850}+z^{500}+1=0$ . Subtracting we have $0=z^{500}-z^{350}=z^{350}(z^{150}-1)=z^{350}(z^{50}-1)(z^{100}+z^{50}+1)$ Now $z^{350}\neq 0$ else $f(z)=1\neq 0$ , and $z^{50}\neq 1$ else $f(z)=3\neq0$ . Hence if $|z|=1$ and $f(z)=0$ then we must have $z^{100}+z^{50}+1=0$ . Conversely, if $z$ satisfies $z^{100}+z^{50}+1=0$ then $z^{150}=1$ so that $|z|=1$ and $z^{850}+z^{350}+1=z^{100}+z^{50}+1=0$ . Therefore $z$ satisfies $f(z)=0$ and $|z|=1$ if and only if $z^{100}+z^{50}+1=0$ . Note that this equation has 100 solutions (by the fundamental theorem of algebra) lying on the unit circle. Furthermore they are distinct since the solutions of $z^{150}=1$ are distinct and $z^{100}+z^{50}+1$ is a factor of $z^{150}-1$ . Therefore the answer is 100.

Solution 3

We have $\frac{z^{850}+z^{350}+1}{3}=0$ . Geometrically speaking, this means that the centroid with vertices on $z^{850}, z^{350}, 1$ is the origin. However, because the circumcenter is also the origin, the only configuration that works is if all three points form an equilateral triangle. Thus, $z^{850}, z^{350}$ are the primitive 3rd roots of unity in some order. Let $y=z^50$ , then because $17$ and $7$ are relatively prime, there is one solution of $y$ for each permutation, so $2$ in total. Thus, there are $2*50=100$ solutions.

@@ Line 2: / Line 2: @@
 Find the number of complex numbers <math>z</math> satisfying <math>|z|=1</math> and <math>z^{850}+z^{350}+1=0</math>.
-==Solution==
+==Solution 1==
-We reduce the problem to <math>z^17+z^7+1</math>, remembering to multiply the final product by 50. We need the imaginary parts of the numbers <math>z^17,z^7</math> to cancel, which by working modulo 360 we can easily determine only happens when the number is of the form <math>\cis(15x)</math>(this holds true because we are only looking for solutions with a magnitude of 1). We also need the real parts to sum to -1. We check all the multiples of 15 that result in <math>\cis(x)</math> being negative, and find that only two work(or alternatively, if you are good, you can guess that only 120 and 240 work). The answer is then 100.
+We reduce the problem to <math>z^{17}+z^7+1</math>, remembering to multiply the final product by 50. We need the imaginary parts of the numbers <math>z^{17},z^7</math> to cancel, which by working modulo 360 we can easily determine only happens when the number is of the form <math>cis(15x)</math> (this holds true because we are only looking for solutions with a magnitude of <math>1</math>). We also need the real parts to sum to <math>-1</math>. We check all the multiples of 15 that result in <math>cis(x)</math> being negative, and find that only two work(or alternatively, if you are good, you can guess that only <math>120</math> and <math>240</math> work). The answer is then <math>100</math>.
+===Note===
+One definite way to find <math>cisa</math> and <math>cisb</math> such that they sum to <math>-1</math> is by setting up equations.
+Let cos<math>a=x</math> and cos<math>b=y</math>. By definition
+<cmath>x+y=-1</cmath>
+<cmath>\sqrt{1-x^2}+\sqrt{1-y^2}=0</cmath>
+We can ingnore plus or minus for the sins as this would still yield the same result. From the second equation <math>1-x^2=1-y^2</math> and <math>x= \pm y</math>.
+Plugging into the first equation, we see that <math>x=-1/2, y=-1/2</math>. And we have <math>a=120</math> and <math>b=240</math> or the other way around.
+(Note by Bigbrain_2009)
+==Solution 2==
+Let <math>f(w)=w^{850}+w^{350}+1</math> and suppose <math>z</math> is such that <math>f(z)=0</math> and <math>|z|=1</math>. Note that <math>z</math> is not real (by descartes rule of signs), thus <math>\overline{z}=|z|^2/z=1/z</math> is also a root of <math>f</math>. It follows that <math>z^{850}f(1/z)=z^{850}+z^{500}+1=0</math>. Subtracting we have <cmath>0=z^{500}-z^{350}=z^{350}(z^{150}-1)=z^{350}(z^{50}-1)(z^{100}+z^{50}+1)</cmath>
+Now <math>z^{350}\neq 0</math> else <math>f(z)=1\neq 0</math>, and <math>z^{50}\neq 1</math> else <math>f(z)=3\neq0</math>. Hence if <math>|z|=1</math> and <math>f(z)=0</math> then we must have <math>z^{100}+z^{50}+1=0</math>. Conversely, if <math>z</math> satisfies <math>z^{100}+z^{50}+1=0</math> then <math>z^{150}=1</math> so that <math>|z|=1</math> and <math>z^{850}+z^{350}+1=z^{100}+z^{50}+1=0</math>. Therefore <math>z</math> satisfies <math>f(z)=0</math> and <math>|z|=1</math> if and only if <math>z^{100}+z^{50}+1=0</math>. Note that this equation has 100 solutions (by the fundamental theorem of algebra) lying on the unit circle. Furthermore they are distinct since the solutions of <math>z^{150}=1</math> are distinct and <math>z^{100}+z^{50}+1</math> is a factor of <math>z^{150}-1</math>. Therefore the answer is 100.
+==Solution 3==
+We have <math>\frac{z^{850}+z^{350}+1}{3}=0</math>. Geometrically speaking, this means that the centroid with vertices on <math>z^{850}, z^{350}, 1</math> is the origin. However, because the circumcenter is also the origin, the only configuration that works is if all three points form an equilateral triangle. Thus, <math>z^{850}, z^{350}</math> are the primitive 3rd roots of unity in some order. Let <math>y=z^50</math>, then because <math>17</math> and <math>7</math> are relatively prime, there is one solution of <math>y</math> for each permutation, so <math>2</math> in total. Thus, there are <math>2*50=100</math> solutions.
 ==See also==
 {{CIME box|year=2020|n=I|num-b=5|num-a=7}}
 {{MAC Notice}}

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