1975 USAMO Problems/Problem 4
Problem
Two given circles intersect in two points and . Show how to construct a segment passing through and terminating on the two circles such that is a maximum.
Solution
A maximum cannot be attained if intersects segment because a larger value can be attained by making one of or diametrically opposite , which (as is easily checked) increases the value of both and . Thus, assume does not intersect .
Let and be the centers of the small and big circles, respectively, and and be their respective radii.
Let and be the feet of and to , and and
We have:
is maximum when the product is a maximum.
We have
But and is fixed, so is .
So its maximum depends on which occurs when . To draw the line :
Draw a circle with center and radius to cut the radius at . Draw the line parallel to passing through . This line meets the small and big circles at and , respectively.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
Solution with graph at Cut the Knot
1975 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
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