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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...")
 
(→‎Solution)
Line 3: Line 3:
  
 
== 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 $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$.

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 $O_1$ and $O-2$ be the centers of circles $k_1$ and $k_2$ respectively.

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