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Problems created by: Eric P. Choate, Radford University Andrew Nelson, Radford University Charlene J. Galvez, Radford University Adam Downs, Radford University Carrie S. Case, Radford University Stephen P. Corwin, Radford University HEXA Challenge

Challenge 1:

When 4 is added to two numbers, the ratio is 5:6. When 4 is subtracted from those two numbers, the ratio is 1:2. What are the two numbers?

Challenge 2: Without talking to each other, roommates Bob and John each stopped at the Soup Store on the way home from work. Bob bought three cans of Broth Bros. Tomato Soup and two cans of Souperman ’ s Clam Chowder for a total of $6.12. John also bought three cans of Broth Bros. Tomato Soup, but he bought one can of Souperman ’ s Gumbo for a total of $4.52. Neither kept his receipt. They do not remember the costs of any of these soups, but they know that Broth Bros. cans are cheaper than Souperman ’ s and all Broth Bros. cans are often on sale for a low as $0.98 per can, but today ’ s price could have been higher than that. Assuming prices at the Soup Store do not include fractions of cents, 1. What is the maximum price of one can of Broth Bros. soup? 2. Which is more expensive, one can of Souperman ’ s Clam Chowder or Souperman ’ s Gumbo?

Challenge 3:

Let n ≥ 4 be an integer, and consider the irregular convex n - gon with exterior angles θ , 2 θ , 3 θ , ..., n θ for some angle θ measured in degrees. For what values of n is θ an integer?

Challenge 4: An ellipse and a hyperbola share the same foci (±13, 0), and they intersect at four points. The rectangle with vertices at these four points of intersection has area 240 and perimeter 68. Find the equations of the ellipse and hyperbola.

Virginia Mathematics Teacher vol. 47, no. 1

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