Difference between revisions of "Northeastern WOOTers Mock AIME I Problems/Problem 10"
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Our strategy is to take advantage of degrees of freedom. The given condition appears extremely weak (that is, it offers little information), yet apparently it uniquely determines <math>k^4</math>. Counterintuitively, this very fact offers lots of information. | Our strategy is to take advantage of degrees of freedom. The given condition appears extremely weak (that is, it offers little information), yet apparently it uniquely determines <math>k^4</math>. Counterintuitively, this very fact offers lots of information. | ||
+ | |||
'''Degree of Freedom 1: Translation''' | '''Degree of Freedom 1: Translation''' | ||
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<cmath> a^2 - ac + c^2 = \frac{a^3 + c^3}{a + c} = 0. </cmath> | <cmath> a^2 - ac + c^2 = \frac{a^3 + c^3}{a + c} = 0. </cmath> | ||
Then <math>a^3 = -c^3</math>, with <math>a \neq -c</math>; this implies <math>|a| = |c|</math>. | Then <math>a^3 = -c^3</math>, with <math>a \neq -c</math>; this implies <math>|a| = |c|</math>. | ||
+ | |||
'''Degree of Freedom 2: Dilation''' | '''Degree of Freedom 2: Dilation''' | ||
Observe that replacing <math>a</math>, <math>b</math>, <math>c</math>, with <math>ra</math>, <math>rb</math>, <math>rc</math>, respectively, has no effect on the condition. Then, an appropriate <math>r</math> can be chosen such that <math>|a| = |c| = 1</math>; that is, without loss of generality, <math>|a| = |c| = 1</math>. | Observe that replacing <math>a</math>, <math>b</math>, <math>c</math>, with <math>ra</math>, <math>rb</math>, <math>rc</math>, respectively, has no effect on the condition. Then, an appropriate <math>r</math> can be chosen such that <math>|a| = |c| = 1</math>; that is, without loss of generality, <math>|a| = |c| = 1</math>. | ||
+ | |||
'''Degree of Freedom 3: Rotation''' | '''Degree of Freedom 3: Rotation''' | ||
− | Let's take a closer look at the given condition. We have already changed it into <math>a^3 = -c^3</math>, <math>a \neq -c</math>. Let <math>a = | + | Let's take a closer look at the given condition. We have already changed it into <math>a^3 = -c^3</math>, <math>a \neq -c</math>. Let <math>a = \text{cis} |
+ | (\theta_a)</math> and <math>c = \text{cis} (\theta_c)</math>. By methods such as [[De Moivre's Theorem]], we determine the condition is true if and only if | ||
<cmath> \theta_c = \pm \frac{\pi}{3} - \theta_a </cmath> | <cmath> \theta_c = \pm \frac{\pi}{3} - \theta_a </cmath> | ||
− | Since this relationship supposedly | + | Since this relationship is supposedly enough to fix <math>k^4</math>, we can set <math>\theta_a = 0 \Rightarrow a = 1</math> without loss of generality. |
From here, we determine <math>\theta_c = \pm \frac{\pi}{3}</math> and <math>c = \frac{1}{2} \pm \frac{\sqrt{3}}{2}</math>. Then we can compute | From here, we determine <math>\theta_c = \pm \frac{\pi}{3}</math> and <math>c = \frac{1}{2} \pm \frac{\sqrt{3}}{2}</math>. Then we can compute | ||
<cmath> k = \pm \sqrt{3}i \Rightarrow k^4 = \boxed{009}. </cmath> | <cmath> k = \pm \sqrt{3}i \Rightarrow k^4 = \boxed{009}. </cmath> |
Latest revision as of 14:08, 9 August 2021
Problem 10
If are complex numbers such that
then find the value of
.
Solution
Our strategy is to take advantage of degrees of freedom. The given condition appears extremely weak (that is, it offers little information), yet apparently it uniquely determines . Counterintuitively, this very fact offers lots of information.
Degree of Freedom 1: Translation
Observe that replacing ,
,
with
,
,
, respectively, has no effect on the condition. Then, by setting
, we can set
without loss of generality.
Substituting this into the condition and clearing denominators yields
Then
, with
; this implies
.
Degree of Freedom 2: Dilation
Observe that replacing ,
,
, with
,
,
, respectively, has no effect on the condition. Then, an appropriate
can be chosen such that
; that is, without loss of generality,
.
Degree of Freedom 3: Rotation
Let's take a closer look at the given condition. We have already changed it into ,
. Let
and
. By methods such as De Moivre's Theorem, we determine the condition is true if and only if
Since this relationship is supposedly enough to fix
, we can set
without loss of generality.
From here, we determine and
. Then we can compute