Triangle Rotating in Square

Equilateral triangle EKP with side EK of length 2 inches is placed inside square EAMI with side of length 4 inches so that K is on side EA. The triangle is rotated clockwise about K, then P, and so on along the sides of the square until E, K, and P return to their original positions. The length of the path in inches traversed by vertex P is equal to:

  1. 20/3
  2. 32/3
  3. 12
  4. 40/3
  5. 15

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Cubic to Quadratic to Linear

One root of a certain third-degree equation is 1. When the cubic term of the equation is crossed off, the resulting quadratic equation has a root of 2. When the squared term is also crossed off, the resulting linear equation has a root of 3. Reconstruct the original third-degree equation, expressing it in the form ax3 + bx2 + cx = d, with all coefficients as integers.

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Brick Wall Without Picture

Tim has employed Kathy and Leslie as bricklayers for the summer. Chris wants a brick wall built between his diving pool and the main swimming pool. Tim estimates that Leslie could build the wall in 9 hours and Kathy could build it in 10. However, he has learned that when they work together their combined output decreases by 10 bricks per hour. But, being in a hurry, he puts them both to work on it and finds that it takes them exactly 5 hours, working together, to finish the wall. How many bricks are in the wall?

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Cat Burglar, Crate

It is night. You are in a quiet city, near the sea. There is no moon. Near a wharf, a cubical wooden shipping crate is lying along the outside wall of a warehouse, flat against the wall. Each edge of the crate is 1 meter. At midnight, out of the crate emerges Cathy, the notorious cat burglar. She noiselessly leans a √15 meter ladder against the wall in such a way that it just grazes the free horizontal edge of the crate. At what height does the ladder hit the wall?

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