Difference between revisions of "1982 USAMO Problems"
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==Problem 5== | ==Problem 5== | ||
<math>O</math> is the center of a sphere <math>S</math>. Points <math>A, B, C</math> are inside <math>S</math>, <math>OA</math> is perpendicular to <math>AB</math> and <math>AC</math>, and there are two spheres through <math>A, B</math>, and <math>C</math> which touch <math>S</math>. Show that the sum of their radii equals the radius of <math>S</math>. | <math>O</math> is the center of a sphere <math>S</math>. Points <math>A, B, C</math> are inside <math>S</math>, <math>OA</math> is perpendicular to <math>AB</math> and <math>AC</math>, and there are two spheres through <math>A, B</math>, and <math>C</math> which touch <math>S</math>. Show that the sum of their radii equals the radius of <math>S</math>. | ||
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+ | == See Also == | ||
+ | {{USAMO box|year=1982|before=[[1981 USAMO]]|after=[[1983 USAMO]]}} |
Revision as of 14:35, 17 September 2012
Problem 1
A graph has points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to points?
Problem 2
Show that if are positive integers such that for all real with sum , then or .
Problem 3
is a point inside the equilateral triangle . is a point inside . Show that
Problem 4
Show that there is a positive integer such that, for every positive integer , is composite.
Problem 5
is the center of a sphere . Points are inside , is perpendicular to and , and there are two spheres through , and which touch . Show that the sum of their radii equals the radius of .
See Also
1982 USAMO (Problems • Resources) | ||
Preceded by 1981 USAMO |
Followed by 1983 USAMO | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |