Challenge Problems: Difficulty Level 3

Challenge Problems (Difficulty Level 3) provide a challenge for well rounded high school math students, but are only moderately challenging for seasoned math competitors. 		 
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Challenge (Difficulty Level 3) Math Competition Problems

The problems on this page are Difficulty Level 3 problems written by Douglas Twitchell. These are good competition problems for well rounded high school math students. Some may be solved more easily with a 'flash of insight', but can usually be solved by more pedantic methods. Brief (not complete) solutions are shown in green, leaving the reader to work through the logic.

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3.1
A dog is inside a yard that is 100 feet by 100 feet. He is chained to one wall, at the midpoint of the wall. His chain is 100 feet long. What is the total area the dog is able to access inside the yard?

    Student has to visualize/diagram the region accessible, break it into component areas, and sum the areas. Answer is 5000/3p + 2500sqrt(3)
3.2
In right triangle ABC, angle C is 90 degrees. If the longer leg is decreased by n and the shorter leg is increased by n, the area remains constant. In terms of the area x, and the hypotenuse c, find n.

    This problem requires students to think in abstract algebraic terms, instead of with numbers. The solution is sqrt(c2 - 4x).
3.3
If f(x) = x3 + x2 - x, find all points where f(x) and its inverse intersect.

    Wherever the function f(x) intersects the line y = x, it also intersects its inverse. Following this logic through gives us the following solutions (0,0), (-2,-2), (1,1)