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16. This problem calls for finding the Isoperimetric point. A Cornell student, Elliott Back, posted this answer on his Web site, which looks right to us.
Draw a circle around C of radius (a+b-c)/2. This makes it easy to draw circles around A and B which are tangent to C. You have three small circles--now draw a final circle outside and tangent to circles A, B, C. The center of this circle is P. This is also known as Apollonius' problem.
17. 199981. This requires a computer program to solve, or a lot of patience.
18. If you’re interested in cool hacks, Tara Calishain’s book Google Hacks is a good place to turn. We also like the many cool and useful things being done with Google Maps. See the Google Maps Mania blog for a good collection.
19. The first half of the question is a fancy way of stating the binomial coefficient:
(n, k) = (n, n-k)
An identity which solves the problem is (n-1, k-1) + (n-1, k) = (n, k).
Or, in words, "I pick k-1, you pick k." Thanks, Elliott.
20. Hint: Write out the numbers, one below another.
The answer: Each number, when spelled out, is the largest for that given word size. Ten is the largest number with three letters; nine the largest with four. Thus, the next line in the series is the largest number with nine letters. That number is ninety-six, so the correct answer is A.
21. We’d love to know your ideas as we continue to cover Google and work on future editions of The Google Story.
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