Cream Of The Crop: Difficulty Level 4

Cream Of The Crop (Difficulty Level 4) problems are specifically designed to distinguish between the good math competitors and the exceptional math competitors. 		 
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Cream Of The Crop: Difficulty Level 4 Math Competition Problems

The problems on this page are Difficulty Level 4 problems written by Douglas Twitchell. These are more challenging problems for high school level math competitions. Including Level 4 level problems in your competition will help distinguish between really good and exceptional students. Not all problems have lengthy solutions; some are easy to solve, provided you have a flash of insight. Brief (not complete) solutions are shown in green, leaving the reader to work through the logic.

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4.1
Find the following sum:
1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + ...

    The trick here is to see each fraction as a difference of fractions, rather than a product of fractions. The answer is 1/2
4.2
Two circles (A and B) with radius 3 are externally tangent to the circle x2 - 6x + y2 - 8y - 21 = 0 and also tangent to the line y = 4/3x. If k is the distance between the centers of the two circles A and B, find all possible values of k.

    The solution to this problem is rather lengthy; the challenge is to recognize that there are three ways to arrange the circles that match the given criteria. The answers are 6, 8, and 10.
4.3
Solve the system of equations:
x2 + y2 + x + y = 12
xy + x + y = 3

    One way to solve this problem begins by making the following substitutions in the system: m = x + y, and n = x - y. The solutions are (-3,-3), (3,0), (0,3)