Difference between revisions of "1971 AHSME Problems/Problem 35"
(informed guess solution) |
m (typo fix) |
||
| Line 52: | Line 52: | ||
<math>\boxed{\textbf{(C) }(16+12\sqrt2):1}</math>. | <math>\boxed{\textbf{(C) }(16+12\sqrt2):1}</math>. | ||
| − | == Solution 2 ( | + | == Solution 2 (Informed guess)== |
Draw a good diagram, and especially make sure that you are drawing a right angle. Then, you will realize that the big circle completely dwarfs all of the other circles. Answer choice (C), <math>16+12\sqrt2</math>, is about <math>16+12\cdot1.4=32.8</math>, whereas the greatest of the other answer choices is about <math>6.3</math>, which is unreasonably small based on our diagram. Thus, we choose answer <math>\boxed{\textbf{(C) }(16+12\sqrt2):1}</math>. | Draw a good diagram, and especially make sure that you are drawing a right angle. Then, you will realize that the big circle completely dwarfs all of the other circles. Answer choice (C), <math>16+12\sqrt2</math>, is about <math>16+12\cdot1.4=32.8</math>, whereas the greatest of the other answer choices is about <math>6.3</math>, which is unreasonably small based on our diagram. Thus, we choose answer <math>\boxed{\textbf{(C) }(16+12\sqrt2):1}</math>. | ||
Revision as of 18:49, 8 August 2024
Problem
Each circle in an infinite sequence with decreasing radii is tangent externally to the one following it and to both sides of a given right angle. The ratio of the area of the first circle to the sum of areas of all other circles in the sequence, is
Solution 1
![[asy] import geometry; point A = origin; point B = dir(135); point C = (0,sqrt(2)); point D = dir(45); point X = 1/(1+sqrt(2)/2)*(B-C)+C; point Y = 1/(1+sqrt(2)/2)*(D-C)+C; // Circles draw(circle(A,1)); draw(incircle(triangle(C,X,Y))); // Segments XY and BD draw(X--Y); draw(B--D); // Square draw(A--B--C--D--cycle); // Point Labels dot(A); label("A",A,S); dot(B); label("B",B,W); dot(C); label("C",C,N); dot(D); label("D",D,E); dot(X); label("X",X,NW); dot(Y); label("Y",Y,NE); [/asy]](http://latex.artofproblemsolving.com/2/3/b/23b7b7a692ec894bd229d2364328771c83ee7018.png)
.
Solution 2 (Informed guess)
Draw a good diagram, and especially make sure that you are drawing a right angle. Then, you will realize that the big circle completely dwarfs all of the other circles. Answer choice (C), 
, is about 
, whereas the greatest of the other answer choices is about 
, which is unreasonably small based on our diagram. Thus, we choose answer .
See Also
| 1971 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 34 |
Followed by Last Problem | |
| 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 • 31 • 32 • 33 • 34 • 35 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
