Difference between revisions of "2013 AMC 12A Problems/Problem 19"
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Let circle <math>A</math> intersect <math>AC</math> at <math>D</math> and <math>E</math> as shown. We apply Power of a Point on point <math>C</math> with respect to circle <math>A.</math> This yields the diophantine equation | Let circle <math>A</math> intersect <math>AC</math> at <math>D</math> and <math>E</math> as shown. We apply Power of a Point on point <math>C</math> with respect to circle <math>A.</math> This yields the diophantine equation | ||
− | <cmath>CX \cdot CB = CD \cdot | + | <cmath>CX \cdot CB = CD \cdot CE</cmath> |
<cmath>CX(CX+XB) = (97-86)(97+86)</cmath> | <cmath>CX(CX+XB) = (97-86)(97+86)</cmath> | ||
<cmath>CX(CX+XB) = 3 \cdot 11 \cdot 61.</cmath> | <cmath>CX(CX+XB) = 3 \cdot 11 \cdot 61.</cmath> |
Revision as of 14:24, 19 September 2021
Contents
[hide]Problem
In , , and . A circle with center and radius intersects at points and . Moreover and have integer lengths. What is ?
Solution
Solution 1 (Diophantine PoP)
Let circle intersect at and as shown. We apply Power of a Point on point with respect to circle This yields the diophantine equation
Since lengths cannot be negative, we must have This generates the four solution pairs for :
However, by the Triangle Inequality on we see that This implies that we must have
(Solution by unknown, latex/asy modified majorly by samrocksnature)
Solution 2
Let , , and meet the circle at and , with on . Then . Using the Power of a Point, we get that . We know that , and that by the triangle inequality on . Thus, we get that
Solution 3
Let represent , and let represent . Since the circle goes through and , . Then by Stewart's Theorem,
(Since cannot be equal to , dividing both sides of the equation by is allowed.)
The prime factors of are , , and . Obviously, . In addition, by the Triangle Inequality, , so . Therefore, must equal , and must equal
Video Solution by Richard Rusczyk
https://artofproblemsolving.com/videos/amc/2013amc12a/357
~dolphin7
Video Solution
~sugar_rush
See also
2013 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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