Michigan Mathematics Prize Competition

1. An Egyptian fraction has the form 1/n, where n is a positive integer. In ancient Egypt, these were the only fractions allowed. Other fractions between zero and one were always expressed as a sum of distinct Egyptian fractions. For example, 3/5 was seen as 1/2 + 1/10, or 1/3 + 1/4 + 1/60. The preferred method of representing a fraction in Egypt used the "greedy" algorithm, which at each stage, uses the Egyptian fraction that eats up as much as possible of what is left of the original fraction. Thus the greedy fraction for 3/5 would be 1/2 + 1/10.
   1. Find the greedy Egyptian fraction representations for 2/13.
   2. Find the greedy Egyptian fraction representations for 9/10.
   3. Find the greedy Egyptian fraction representations for 2/(2k+1), where k is a positive integer.
   4. Find the greedy Egyptian fraction representations for 3/(6k+1), where k is a positive integer.
   
   
2. 1. The smaller of two concentric circles has radius one unit. The area of the larger circle is twice the area of the smaller circle. Find the difference in their radii.
   2. The smaller of two identically oriented equilateral triangles has each side one unit long. The smaller triangle is centered within the larger triangle so that the perpendicular distance between parallel sides is always the same number d. The area of the larger triangle is twice the area of the smaller triangle. Find d.
   
   
3. Suppose that the domain of a function f is the set of real numbers and that f takes values in the set of real numbers. A real number x₀ is a fixed point of f if f(x₀) = x₀.
   1. Let f(x) = m x + b. For which m does f have a fixed point?
   2. Find the fixed point of f(x) = m x + b in terms of m and b, when it exists.
   3. Consider the functions f_c(x) = x² - c.
      1. For which values of c are there two different fixed points?
      2. For which values of c are there no fixed points?
      3. In terms of c, find the value(s) of the fixed point(s).
   4. Find an example of a function that has exactly three fixed points.
   
   
4. A square based pyramid is made out of rubber balls. There are 100 balls on the bottom level, 81 on the next level, etc., up to 1 ball on the top level.
   1. How many balls are there in the pyramid?
   2. If each ball has a radius of 1 meter, how tall is the pyramid?
   3. What is the volume of the solid that you create if you place a plane against each of the four sides and the base of the balls?
   
   
5. We wish to consider a general deck of cards specified by a number of suits, a sequence of denominations, and a number (possibly 0) of jokers. The deck will consist of exactly one card of each denomination from each suit, plus the jokers, which are "wild" and can be counted as any possible card of any suit. For example, a standard deck of cards consists of 4 suits, 13 denominations, and 0 jokers.
   1. For a deck with 3 suits {a, b, c} and 7 denominations {1, 2, 3, 4, 5, 6, 7}, and 0 jokers, find the probability that a 3-card hand will be a straight. (A straight consists of 3 cards in sequence, e.g., {a1, c3, a2} but not {a6, b7, c1}).
   2. For a deck with 3 suits, 7 denominations, and 0 jokers, find the probability that a 3-card hand will consist of 3 cards of the same suit (i.e., a flush).
   3. For a deck with 3 suits, 7 denominations, and 1 joker, find the probability that a 3-card hand dealt at random will be a straight and also the probability that a 3-card hand will be a flush.
   4. Find a number of suits and the length of the denomination sequence that would be required if a deck is to contain 1 joker and is to have identical probabilities for a straight and a flush when a 3-card hand is dealt. The answer that you find must be an answer such that a flush and a straight are possible but not certain to occur.

Modified August 14, 1999, by Robert Messer