Difference between revisions of "2008 AIME II Problems/Problem 13"
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We note that if the region <math>S_2 = \left\lbrace\frac{1}{z}|z \in R_2\right\rbrace</math>, where <math>R_2</math> is the region (in green below) outside the circle of radius <math>1/\sqrt{3}</math> centered at the origin, then <math>S_2</math> is simply the region inside a circle of radius <math>\sqrt{3}</math> centered at the origin. It now suffices to find what happens to the mapping of the region <math>R_2 - R</math> (in blue below). | We note that if the region <math>S_2 = \left\lbrace\frac{1}{z}|z \in R_2\right\rbrace</math>, where <math>R_2</math> is the region (in green below) outside the circle of radius <math>1/\sqrt{3}</math> centered at the origin, then <math>S_2</math> is simply the region inside a circle of radius <math>\sqrt{3}</math> centered at the origin. It now suffices to find what happens to the mapping of the region <math>R_2 - R</math> (in blue below). | ||
− | The equation of the hexagon side in that region is <math>x = r \cos \theta = \frac{1}{2}</math>, which is transformed to <math>\frac{1}{r} \cos (-\theta) = \frac{1}{r} \cos \theta = 2. Let </math>r\cos \theta = a+bi<math> where </math>a,b \in \mathbb{R}<math>; then </math>r = \sqrt{a^2 + b^2}, \cos \theta = \frac{a}{\sqrt{a^2 + b^2}}<math>, so the equation becomes </math>a^2 - 2a + b^2 = 0 \Longrightarrow (a-1)^2 + b^2 = 1<math>. Hence the side is sent to an arc of the unit circle centered at </math>(1,0)<math>, after considering the restriction that the side of the hexagon is a segment of length </math>1/\sqrt{3}<math>. | + | The equation of the hexagon side in that region is <math>x = r \cos \theta = \frac{1}{2}</math>, which is transformed to <math>\frac{1}{r} \cos (-\theta) = \frac{1}{r} \cos \theta = </math>2<math>. Let </math>r\cos \theta = a+bi<math> where </math>a,b \in \mathbb{R}<math>; then </math>r = \sqrt{a^2 + b^2}, \cos \theta = \frac{a}{\sqrt{a^2 + b^2}}<math>, so the equation becomes </math>a^2 - 2a + b^2 = 0 \Longrightarrow (a-1)^2 + b^2 = 1<math>. Hence the side is sent to an arc of the unit circle centered at </math>(1,0)<math>, after considering the restriction that the side of the hexagon is a segment of length </math>1/\sqrt{3}<math>. |
Including </math>S_2<math>, we find that </math>S<math> is the union of six unit circles centered at </math>\text{cis}\, \frac{k\pi}{6}<math>, </math>k = 0,1,2,3,4,5$, as shown below. | Including </math>S_2<math>, we find that </math>S<math> is the union of six unit circles centered at </math>\text{cis}\, \frac{k\pi}{6}<math>, </math>k = 0,1,2,3,4,5$, as shown below. |
Revision as of 11:21, 1 March 2016
Problem
A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let be the region outside the hexagon, and let . Then the area of has the form , where and are positive integers. Find .
Solution 1
If a point is in , then the point is in (where cis denotes ). Since is symmetric every about the origin, it suffices to consider the area of the result of the transformation when , and then to multiply by to account for the entire area.
We note that if the region , where is the region (in green below) outside the circle of radius centered at the origin, then is simply the region inside a circle of radius centered at the origin. It now suffices to find what happens to the mapping of the region (in blue below).
The equation of the hexagon side in that region is , which is transformed to 2r\cos \theta = a+bia,b \in \mathbb{R}r = \sqrt{a^2 + b^2}, \cos \theta = \frac{a}{\sqrt{a^2 + b^2}}a^2 - 2a + b^2 = 0 \Longrightarrow (a-1)^2 + b^2 = 1(1,0)1/\sqrt{3}$.
Including$ (Error compiling LaTeX. ! Missing $ inserted.)S_2S\text{cis}\, \frac{k\pi}{6}k = 0,1,2,3,4,5$, as shown below.
The area of the regular hexagon is . The total area of the six sectors is . Their sum is , and .
Solution 2 (Calculus)
One can describe the line parallel to the imaginary axis using polar coordinates as
so is equal to
Dividing the hexagon to 12 equal parts we get that
which is a routine computation:
.
See also
2008 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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