1. Find all solutions in integers of x3 + 2y3 = 4z3. (p. 45)

2. The Wohascum County Board of Commissioners, which has 20 members, recently had to elect a President. There were three candidates (A, B, and C); on each ballot the three candidates were to be listed in order of preference, with no abstentions. It was found that 11 members, a majority, preferred A over B (thus the other 9 preferred B over A). Similarly, it was found that 12 members preferred C over A. Given these results, it was suggested that B should withdraw, to enable a runoff election between A and C. However, B protested, and it was then found that 14 members preferred B over C! The Board has not yet recovered from the resulting confusion. Given that every possible order of A, B, C appeared on at least one ballot, how many board members voted for B as their first choice? (p. 46)

3. If A = (0,−10) and B = (2, 0), find the point(s) C on the parabola y = x2 which minimizes the area of triangle ABC. (p. 47)

4. Does there exist a continuous function y = f(x), defined for all real x, whose graph intersects every non-vertical line in infinitely many points? (Note that because f is a function, its graph will intersect every vertical line in exactly one point.) (p. 48)

5. A child on a pogo stick jumps 1 foot on the first jump, 2 feet on the second jump, 4 feet on the third jump, …, 2n−1 feet on the nth jump. Can the child get back to the starting point by a judicious choice of directions? (p. 49)