# Difference between revisions of "2012 JBMO Problems/Problem 2"

(Created page with "== Section 2== Let the circles <math>k_1</math> and <math>k_2</math> intersect at two points <math>A</math> and <math>B</math>, and let <math>t</math> be a common tangent of...") |
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== Solution == | == Solution == | ||

+ | <asy> | ||

+ | size(15cm,0); | ||

+ | draw((0,0)--(0,2)--(4,2)--(4,-3)--(0,0)); | ||

+ | draw((-1,2)--(9,2)); | ||

+ | draw((0,0)--(2,2)); | ||

+ | draw((2,2)--(1,1)); | ||

+ | draw(circle((0,1),1)); | ||

+ | draw(circle((4,-3),5)); | ||

+ | dot((0,0)); | ||

+ | dot((0,2)); | ||

+ | dot((2,2)); | ||

+ | dot((4,2)); | ||

+ | dot((4,-3)); | ||

+ | dot((1,1)); | ||

+ | dot((0,1)); | ||

+ | label("A",(0,0),NW); | ||

+ | label("B",(1,1),NW); | ||

+ | label("M",(0,2),N); | ||

+ | label("N",(4,2),N); | ||

+ | label("$O_1$",(0,1),NW); | ||

+ | label("$O_2$",(4,-3),NE); | ||

+ | label("$k_1$",(-0.7,1.7),NW); | ||

+ | label("$k_2$",(7.6,0.46),NE); | ||

+ | label("$t$",(7.5,2),N); | ||

+ | label("P",(2,2),N); | ||

+ | </asy> | ||

+ | |||

+ | Let <math>O_1</math> and <math>O-2</math> be the centers of circles <math>k_1</math> and <math>k_2</math> respectively. |

## Revision as of 21:16, 22 December 2020

## Section 2

Let the circles and intersect at two points and , and let be a common tangent of and that touches and at and respectively. If and , evaluate the angle .

## Solution

Let and be the centers of circles and respectively.