Difference between revisions of "1976 AHSME Problems/Problem 24"
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/* Made by Klaus-Anton, Edited by MRENTHUSIASM */ | /* Made by Klaus-Anton, Edited by MRENTHUSIASM */ | ||
size(200); | size(200); | ||
| − | pair K=(0,0),B=(1,0),A=(-1,0),L=(0,0.5),M=(sqrt(2)/2,.25),I=(2*sqrt(2)/3,1/3),E=(sqrt(2)/3,1/3); | + | pair K=(0,0),B=(1,0),A=(-1,0),L=(0,0.5),M=(sqrt(2)/2,.25),I=(2*sqrt(2)/3,1/3),E=(sqrt(2)/3,1/3),P=(0,0.25); |
draw(circle(K,1)^^A--B); | draw(circle(K,1)^^A--B); | ||
draw(circle(L,0.5)^^circle(M,.25)); | draw(circle(L,0.5)^^circle(M,.25)); | ||
| − | label("$A$", A, | + | draw(L--K,red); |
| − | label("$K$", K, | + | draw(L--M,red); |
| − | label("$B$", B, | + | draw(K--I,red); |
| + | draw(P--M,red); | ||
| + | label("$A$", A, (-5/4,0)); | ||
| + | label("$K$", K, (0,-5/4)); | ||
| + | label("$B$", B, (5/4,0)); | ||
label("$L$", L, (0,5/4)); | label("$L$", L, (0,5/4)); | ||
label("$M$", M, (0,5/4)); | label("$M$", M, (0,5/4)); | ||
| + | label("$P$", P, (-5/4,0)); | ||
dot(K,linewidth(4)); | dot(K,linewidth(4)); | ||
dot(L,linewidth(4)); | dot(L,linewidth(4)); | ||
dot(M,linewidth(4)); | dot(M,linewidth(4)); | ||
dot(I,linewidth(4)); | dot(I,linewidth(4)); | ||
| + | dot(E,linewidth(4)); | ||
| + | dot(P,linewidth(4)); | ||
</asy> | </asy> | ||
| − | |||
== See Also == | == See Also == | ||
{{AHSME box|year=1976|num-b=23|num-a=25}} | {{AHSME box|year=1976|num-b=23|num-a=25}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 07:04, 6 September 2021
Problem
In the adjoining figure, circle 
has diameter 
; circle 
is tangent to circle and to at the center of circle ; and circle 
tangent to circle , to circle and . The ratio of the area of circle to the area of circle is
![[asy] /* Made by Klaus-Anton, Edited by MRENTHUSIASM */ size(150); pair K=(0,0),B=(1,0),A=(-1,0),L=(0,0.5),M=(sqrt(2)/2,.25); draw(circle(K,1)^^A--B); draw(circle(L,0.5)^^circle(M,.25)); label("$A$", A, W); label("$K$", K, S); label("$B$", B, E); label("$L$", L); label("$M$", M); [/asy]](http://latex.artofproblemsolving.com/7/b/0/7b0c55f990d80458f68a73509b9da6016ca29e1b.png)
Solution
Let 
and 
be the radii of circles and 
respectively.
![[asy] /* Made by Klaus-Anton, Edited by MRENTHUSIASM */ size(200); pair K=(0,0),B=(1,0),A=(-1,0),L=(0,0.5),M=(sqrt(2)/2,.25),I=(2*sqrt(2)/3,1/3),E=(sqrt(2)/3,1/3),P=(0,0.25); draw(circle(K,1)^^A--B); draw(circle(L,0.5)^^circle(M,.25)); draw(L--K,red); draw(L--M,red); draw(K--I,red); draw(P--M,red); label("$A$", A, (-5/4,0)); label("$K$", K, (0,-5/4)); label("$B$", B, (5/4,0)); label("$L$", L, (0,5/4)); label("$M$", M, (0,5/4)); label("$P$", P, (-5/4,0)); dot(K,linewidth(4)); dot(L,linewidth(4)); dot(M,linewidth(4)); dot(I,linewidth(4)); dot(E,linewidth(4)); dot(P,linewidth(4)); [/asy]](http://latex.artofproblemsolving.com/6/f/7/6f7737b24def11fe290b8c86de915476c2e1efa9.png)
See Also
| 1976 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 23 |
Followed by Problem 25 | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
