Difference between revisions of "1971 AHSME Problems/Problem 35"
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== Solution == | == Solution == | ||
+ | |||
+ | <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> | ||
+ | |||
<math>\boxed{\textbf{(C) }(16+12\sqrt2):1}</math>. | <math>\boxed{\textbf{(C) }(16+12\sqrt2):1}</math>. | ||
Revision as of 18:38, 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
.
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.