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Bailey Challenge For the past 40 years Rose has sponsored a high school mathematics contest with 1200 to 1500 participants each year. The test sites are in St. Louis, Cincinnati and at Rose. I try to find some problems that all can solve and some that none can solve. The triple bonus (a) for this issue of Echoes turned out to be one of the most difficult of the 2006 contest. Problem 1 below was recently posed on National Public Radio.
 Problem 1
The digits 2, 7, and 1, can be combined to give 8 by adding the 2 and the 7 and subtracting the 1. If d is any of the ten digits (0 through 9), then you must combine 2, 7 and d to give 8 with two restrictions:
- 2, 7 and d are to be used once and only once for each combination.
- You may add, subtract, multiply, divide, raise to a power, enclose in parentheses, and use the decimal point (any of these more than once if needed)
Partial credit is awarded for solving eight of nine of the ten parts.
 Triple Bonus
A circle of radius one is inscribed in a square. A small circle is tangent to two sides of the square and to the larger circle as shown.
- Find the exact radius of the smaller circle using geometry and algebra.
- Find the radius of the smaller circle using geometry and trigonometry. If you remember trig identities, then you can get the exact solution found in part a.
- How could the small circle be constructed using only compass and straightedge (Euclid and high school geometry)
Full bonus credit for solving any one of the three parts. Triple bonus for all three.
Send your solutions to Herb.Bailey@rose-hulman.edu or to Herb Bailey, Math. Dept., Rose-Hulman, 5500 Wabash Ave., Terre Haute IN 47803.
PLEASE include your class year if you are an alum.
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