1969 AHSME Problems/Problem 28
Problem
Let be the number of points interior to the region bounded by a circle with radius , such that the sum of squares of the distances from to the endpoints of a given diameter is . Then is:
Solution
Let and be points on diameter. Extend , and mark intersection with circle as point .
Because is a diameter, . Also, by Exterior Angle Theorem, , so , making an obtuse angle.
By the Law of Cosines, . Since , substitute and simplify to get . This equation has infinite solutions because for every and , where and and are both less than , there can be an obtuse angle that satisfies the equation, so the answer is .
See also
1969 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 27 |
Followed by Problem 29 | |
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