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Difference between revisions of "Mock AIME 1 Pre 2005 Problems/Problem 10"

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== Problem ==
 
== Problem ==
 
<math>ABCDEFG</math> is a regular heptagon inscribed in a unit circle centered at <math>O</math>. <math>l</math> is the line tangent to the circumcircle of <math>ABCDEFG</math> at <math>A</math>, and <math>P</math> is a point on <math>l</math> such that triangle <math>AOP</math> is isosceles. Let <math>p</math> denote the value of <math>AP \cdot BP \cdot CP \cdot DP \cdot EP \cdot FP \cdot GP</math>. Determine the value of <math>p^2</math>.
 
<math>ABCDEFG</math> is a regular heptagon inscribed in a unit circle centered at <math>O</math>. <math>l</math> is the line tangent to the circumcircle of <math>ABCDEFG</math> at <math>A</math>, and <math>P</math> is a point on <math>l</math> such that triangle <math>AOP</math> is isosceles. Let <math>p</math> denote the value of <math>AP \cdot BP \cdot CP \cdot DP \cdot EP \cdot FP \cdot GP</math>. Determine the value of <math>p^2</math>.
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== Solution ==
 
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Let in the complex plane <math>P = 0+i0</math>, <math>A = 0+i</math>, <math>O = -1+i</math>. Then the vertices of our hexagon are at points <math>z_1, \cdots , z_7</math>, where <math>z_k - (-1+i)</math> are the 7th roots of unity, ie. the complex roots of <math>f(z) = (z-(-1+i))^7=1</math>. If <math>z_k = a_k+ib_k</math>, then what we want is <math>p^2 = \prod_{i=1}^7 (a_k^2+ib_k^2) = \prod_{i=1}^7 (a_k+b_k)(a_k-b_k) = \prod_{i=1}^7 z_k\cdot\bar{z_k}</math>. Notice that <math>\bar{z_k}-(-1+-i)</math> are also the 7th roots of unity, ie. the complex roots of <math>g(z) = (z_k -(-1-i))^7=1</math>. From Vieta, <math>\prod_{i=1}^7 z_k</math> is the constant term of <math>f(z)</math>, or <math>f(0) = 7+i8</math> and <math>\prod_{i=1}^7 \bar{z_k}</math> is <math>g(0) = 7-i8</math>. Thus, <math>p^2 = (7-i8)(7+i8) = 7^2+8^2 = 49+64 = \boxed{113}</math>.
 
  
 
== See also ==
 
== See also ==
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{{Mock AIME box|year=Pre 2005|n=1|num-b=8|num-a=10|source=14769}}
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{{Mock AIME box|year=Pre 2005|n=1|num-b=9|num-a=11|source=14769}}
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[[Category:Intermediate Complex Number Problems]]
 

Latest revision as of 23:28, 22 December 2023

Problem

$ABCDEFG$ is a regular heptagon inscribed in a unit circle centered at $O$. $l$ is the line tangent to the circumcircle of $ABCDEFG$ at $A$, and $P$ is a point on $l$ such that triangle $AOP$ is isosceles. Let $p$ denote the value of $AP \cdot BP \cdot CP \cdot DP \cdot EP \cdot FP \cdot GP$. Determine the value of $p^2$.

See also

Mock AIME 1 Pre 2005 (Problems, Source)
Preceded by
Problem 9 		Followed by
Problem 11
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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