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Difference between revisions of "2014 AIME I Problems/Problem 7"

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== Problem 7 ==
 
== Problem 7 ==
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Let <math>w</math> and <math>z</math> be complex numbers such that <math>|w| = 1</math> and <math>|z| = 10</math>. Let <math>\theta = \arg \left(\tfrac{w-z}{z}\right) </math>. The maximum possible value of <math>\tan^2 \theta</math> can be written as <math>\tfrac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. (Note that <math>\arg(w)</math>, for <math>w \neq 0</math>, denotes the measure of the angle that the ray from <math>0</math> to <math>w</math> makes with the positive real axis in the complex plane.
+
Let <math>w</math> and <math>z</math> be complex numbers such that <math>|w| = 1</math> and <math>|z| = 10</math>. Let <math>\theta = \arg \left(\tfrac{w-z}{z}\right) </math>. The maximum possible value of <math>\tan^2 \theta</math> can be written as <math>\tfrac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. (Note that <math>\arg(w)</math>, for <math>w \neq 0</math>, denotes the measure of the angle that the ray from <math>0</math> to <math>w</math> makes with the positive real axis in the complex plane)
  
−
== Solution ==
+
== Solution 1 ==
−
Let <math>w = \mathrm{cis}{(\alpha)}</math> and <math>z = 10\mathrm{cis}{(\beta)}</math>. Then, <math>\dfrac{w - z}{z} = \dfrac{\mathrm{cis}{(\alpha)} - 10\mathrm{cis}{(\beta)}}{10\mathrm{cis}{\beta}}</math>.
+
Let <math>w = \operatorname{cis}{(\alpha)}</math> and <math>z = 10\operatorname{cis}{(\beta)}</math>. Then, <math>\dfrac{w - z}{z} = \dfrac{\operatorname{cis}{(\alpha)} - 10\operatorname{cis}{(\beta)}}{10\operatorname{cis}{\beta}}</math>.
  
−
Multiplying both the numerator and denominator of this fraction by <math>\mathrm{cis}{(-\beta)}</math> gives us:
+
Multiplying both the numerator and denominator of this fraction by <math>\operatorname{cis}{(-\beta)}</math> gives us:
  
−
<math>\dfrac{w - z}{z} = \dfrac{1}{10}\mathrm{cis}{(\alpha - \beta)} - 1 = \dfrac{1}{10}\mathrm{cos}{(\alpha - \beta)} + \dfrac{1}{10}i\mathrm{sin}{(\alpha - \beta)} - 1</math>.
+
<math>\dfrac{w - z}{z} = \dfrac{1}{10}\operatorname{cis}{(\alpha - \beta)} - 1 = \dfrac{1}{10}\cos{(\alpha - \beta)} + \dfrac{1}{10}i\sin{(\alpha - \beta)} - 1</math>.
  
−
We know that <math>\mathrm{tan}{\theta}</math> is equal to the imaginary part of the above expression divided by the real part. Let <math>x = \alpha - \beta</math>. Then, we have that:
+
We know that <math>\tan{\theta}</math> is equal to the imaginary part of the above expression divided by the real part. Let <math>x = \alpha - \beta</math>. Then, we have that:
  
−
<math>\mathrm{tan}{\theta} = \dfrac{\mathrm{sin}{x}}{\mathrm{cos}{x} - 10}.</math>
+
<math>\tan{\theta} = \dfrac{\sin{x}}{\cos{x} - 10}.</math>
  
 
We need to find a maximum of this expression, so we take the derivative:
 
We need to find a maximum of this expression, so we take the derivative:
  
−
<math>\dfrac{d}{dx} \left (\dfrac{\mathrm{sin}{x}}{\mathrm{cos}{x} - 10} \right) = \dfrac{1 - 10\mathrm{cos}{x}}{(\mathrm{cos}{x} - 10)^2}</math>
+
Note (not from author): To take the derivative, we need to use the [https://en.wikipedia.org/wiki/Quotient_rule Quotient Rule]. In this case, <cmath>\frac{d}{dx}\left(\frac{\sin x}{\cos x-10}\right)=\frac{\cos x(\cos x-10)-(-\sin x)\sin x}{(\cos x-10)^2}=\dfrac{1 - 10\cos{x}}{(\cos{x} - 10)^2}</cmath>
  
−
Thus, we see that the maximum occurs when <math>\mathrm{cos}{x} = \dfrac{1}{10}</math>. Therefore, <math>\mathrm{sin}{x} = \pm\dfrac{\sqrt{99}}{10}</math>, and <math>\mathrm{tan}{\theta} = \pm\dfrac{\sqrt{99}}{99}</math>. Thus, the maximum value of <math>\mathrm{tan^2}{\theta}</math> is <math>\dfrac{99}{99^2}</math>, or <math>\dfrac{1}{99}</math>, and our answer is <math>1 + 99 = \boxed{100}</math>.
+
Thus, we see that the maximum occurs when <math>\cos{x} = \dfrac{1}{10}</math>. Therefore, <math>\sin{x} = \pm\dfrac{\sqrt{99}}{10}</math>, and <math>\tan{\theta} = \pm\dfrac{\sqrt{99}}{99}</math>. Thus, the maximum value of <math>\tan^2{\theta}</math> is <math>\dfrac{99}{99^2}</math>, or <math>\dfrac{1}{99}</math>, and our answer is <math>1 + 99 = \boxed{100}</math>.
  
−
== Solution 2 (No calculus) ==
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== Solution 2 ==
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Without the loss of generality one can let <math>z</math> lie on the positive x axis and since <math>arg(\theta)</math> is a measure of the angle if <math>z=10</math> then <math>arg(\dfrac{w-z}{z})=arg(w-z)</math> and we can see that the question is equivelent to having a triangle <math>OAB</math> with sides <math>OA =10</math> <math>AB=1</math> and <math>OB=t</math> and trying to maximize the angle <math>BOA</math>
+
Without the loss of generality one can let <math>z</math> lie on the positive x axis and since <math>arg(\theta)</math> is a measure of the angle if <math>z=10</math> then <math>arg(\dfrac{w-z}{z})=arg(w-z)</math> and we can see that the question is equivalent to having a triangle <math>OAB</math> with sides <math>OA =10</math> <math>AB=1</math> and <math>OB=t</math> and trying to maximize the angle <math>BOA</math>
 
<asy>
 
<asy>
 
pair O = (0,0);
 
pair O = (0,0);
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dot("$B$",B,dir(45));
 
dot("$B$",B,dir(45));
 
dot("$O$",O,dir(135));
 
dot("$O$",O,dir(135));
−
dot("$ \theta$",O,dir(30));
+
dot("$ \theta$",O,(7,1.2));
  
 
label("$1$", ( A--B ));
 
label("$1$", ( A--B ));
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</asy>
 
</asy>
  
−
using the law of cosines we get:
+
using the Law of Cosines we get:
 
<math>1^2=10^2+t^2-t*10*2\cos\theta</math>
 
<math>1^2=10^2+t^2-t*10*2\cos\theta</math>
 
rearranging:
 
rearranging:
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<cmath>\frac{99}{20t}+\frac{t}{20}=\cos\theta</cmath>
 
<cmath>\frac{99}{20t}+\frac{t}{20}=\cos\theta</cmath>
 
if we want to maximize <math>\theta</math> we need to minimize <math>\cos\theta</math>
 
if we want to maximize <math>\theta</math> we need to minimize <math>\cos\theta</math>
−
, using AM-GM inequality we get that the minimum value for <math>\cos\theta= 2(\sqrt{\dfrac{99}{20t}\dfrac{t}{20}})=2\sqrt{\dfrac{99}{400}}=\dfrac{\sqrt{99}}{10}</math>
+
, using AM-GM inequality we get that the minimum value for <math>\cos\theta= 2\left(\sqrt{\dfrac{99}{20t}\dfrac{t}{20}}\right)=2\sqrt{\dfrac{99}{400}}=\dfrac{\sqrt{99}}{10}</math>
 
hence using the identity <math>\tan^2\theta=\sec^2\theta-1</math>
 
hence using the identity <math>\tan^2\theta=\sec^2\theta-1</math>
 
we get <math>\tan^2\theta=\frac{1}{99}</math>and our answer is <math>1 + 99 = \boxed{100}</math>.
 
we get <math>\tan^2\theta=\frac{1}{99}</math>and our answer is <math>1 + 99 = \boxed{100}</math>.
 +
 +
Note : You can also realize that the max <math>\theta</math> is when the line from <math>0</math> is tangent to the circle of radius <math>1</math> centered at <math>10.</math>
 +
 +
== Solution 3 ==
 +
 +
Note that <math>\frac{w-z}{z}=\frac{w}{z}-1</math>, and that <math>\left|\frac{w}{z}\right|=\frac{1}{10}</math>. Thus <math>\frac{w}{z}-1</math> is a complex number on the circle with radius <math>\frac{1}{10}</math> and centered at <math>-1</math> on the complex plane. Let <math>\omega</math> denote this circle.
 +
 +
Let <math>A</math> and <math>C</math> be the points that represent <math>\frac{w}{z}-1</math> and <math>-1</math> respectively on the complex plane. Let <math>O</math> be the origin. In order to maximize <math>\tan^2(\theta)</math>, we need to maximize <math>\angle{AOC}</math>. This angle is maximized when <math>AO</math> is tangent to <math>\omega</math>. Using the Pythagorean Theorem, we get
 +
 +
<cmath>AO^2=1^2-\left(\frac{1}{10}\right)^2=\frac{99}{100}</cmath>
 +
 +
Thus
 +
 +
<cmath>\tan^2(\theta)=\frac{AC^2}{AO^2}=\frac{1/100}{99/100}=\frac{1}{99}</cmath>
 +
 +
And the answer is <math>1+99=\boxed{100}</math>.
 +
 +
==Video Solution by Punxsutawney Phil==
 +
 +
https://www.youtube.com/watch?v=yakhEuPy6Sg
 +
 +
~sugar_rush
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2014|n=I|num-b=6|num-a=8}}
 
{{AIME box|year=2014|n=I|num-b=6|num-a=8}}
 +
 +
[[Category:Complex numbers]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 19:25, 10 October 2023

Problem 7

Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$. Let $\theta = \arg \left(\tfrac{w-z}{z}\right)$. The maximum possible value of $\tan^2 \theta$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. (Note that $\arg(w)$, for $w \neq 0$, denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane)

Solution 1

Let $w = \operatorname{cis}{(\alpha)}$ and $z = 10\operatorname{cis}{(\beta)}$. Then, $\dfrac{w - z}{z} = \dfrac{\operatorname{cis}{(\alpha)} - 10\operatorname{cis}{(\beta)}}{10\operatorname{cis}{\beta}}$.

Multiplying both the numerator and denominator of this fraction by $\operatorname{cis}{(-\beta)}$ gives us:

$\dfrac{w - z}{z} = \dfrac{1}{10}\operatorname{cis}{(\alpha - \beta)} - 1 = \dfrac{1}{10}\cos{(\alpha - \beta)} + \dfrac{1}{10}i\sin{(\alpha - \beta)} - 1$.

We know that $\tan{\theta}$ is equal to the imaginary part of the above expression divided by the real part. Let $x = \alpha - \beta$. Then, we have that:

$\tan{\theta} = \dfrac{\sin{x}}{\cos{x} - 10}.$

We need to find a maximum of this expression, so we take the derivative:

Note (not from author): To take the derivative, we need to use the Quotient Rule. In this case, \[\frac{d}{dx}\left(\frac{\sin x}{\cos x-10}\right)=\frac{\cos x(\cos x-10)-(-\sin x)\sin x}{(\cos x-10)^2}=\dfrac{1 - 10\cos{x}}{(\cos{x} - 10)^2}\]

Thus, we see that the maximum occurs when $\cos{x} = \dfrac{1}{10}$. Therefore, $\sin{x} = \pm\dfrac{\sqrt{99}}{10}$, and $\tan{\theta} = \pm\dfrac{\sqrt{99}}{99}$. Thus, the maximum value of $\tan^2{\theta}$ is $\dfrac{99}{99^2}$, or $\dfrac{1}{99}$, and our answer is $1 + 99 = \boxed{100}$.

Solution 2

Without the loss of generality one can let $z$ lie on the positive x axis and since $arg(\theta)$ is a measure of the angle if $z=10$ then $arg(\dfrac{w-z}{z})=arg(w-z)$ and we can see that the question is equivalent to having a triangle $OAB$ with sides $OA =10$ $AB=1$ and $OB=t$ and trying to maximize the angle $BOA$ [asy] pair O = (0,0); pair A = (100,0); pair B = (80,30); pair D = (sqrt(850),sqrt(850)); draw(A--B--O--cycle); dotfactor = 3; dot("$A$",A,dir(45)); dot("$B$",B,dir(45)); dot("$O$",O,dir(135)); dot("$ \theta$",O,(7,1.2)); label("$1$", ( A--B )); label("$10$",(O--A)); label("$t$",(O--B)); [/asy]

using the Law of Cosines we get: $1^2=10^2+t^2-t*10*2\cos\theta$ rearranging: \[20t\cos\theta=t^2+99\] solving for $\cos\theta$ we get:

\[\frac{99}{20t}+\frac{t}{20}=\cos\theta\] if we want to maximize $\theta$ we need to minimize $\cos\theta$ , using AM-GM inequality we get that the minimum value for $\cos\theta= 2\left(\sqrt{\dfrac{99}{20t}\dfrac{t}{20}}\right)=2\sqrt{\dfrac{99}{400}}=\dfrac{\sqrt{99}}{10}$ hence using the identity $\tan^2\theta=\sec^2\theta-1$ we get $\tan^2\theta=\frac{1}{99}$and our answer is $1 + 99 = \boxed{100}$.

Note : You can also realize that the max $\theta$ is when the line from $0$ is tangent to the circle of radius $1$ centered at $10.$

Solution 3

Note that $\frac{w-z}{z}=\frac{w}{z}-1$, and that $\left|\frac{w}{z}\right|=\frac{1}{10}$. Thus $\frac{w}{z}-1$ is a complex number on the circle with radius $\frac{1}{10}$ and centered at $-1$ on the complex plane. Let $\omega$ denote this circle.

Let $A$ and $C$ be the points that represent $\frac{w}{z}-1$ and $-1$ respectively on the complex plane. Let $O$ be the origin. In order to maximize $\tan^2(\theta)$, we need to maximize $\angle{AOC}$. This angle is maximized when $AO$ is tangent to $\omega$. Using the Pythagorean Theorem, we get

\[AO^2=1^2-\left(\frac{1}{10}\right)^2=\frac{99}{100}\]

Thus

\[\tan^2(\theta)=\frac{AC^2}{AO^2}=\frac{1/100}{99/100}=\frac{1}{99}\]

And the answer is $1+99=\boxed{100}$.

Video Solution by Punxsutawney Phil

https://www.youtube.com/watch?v=yakhEuPy6Sg

~sugar_rush

See also

2014 AIME I (Problems • Answer Key • Resources)
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Problem 8
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