48th MMPC Part II Problems

The top 985 students had 100 minutes to solve these five problems.
  1. The following figure represents a rectangular piece of paper ABCD whose dimensions are 4 inches by 3 inches. When the paper is folded along the line segment EF, the corners A and C coincide.
    1. Find the length of segment EF.
    2. Extend AD and EF so they meet at G. Find the area of the triangle AEG.
    1. Let p be a prime number. If a, b, c, and d are distinct integers such that the equation
      (x – a) (x – b) (x – c) (x – d) – p2 = 0
      has an integer solution r, show that
      (r – a) + (r – b) + (r – c) + (r – d) = 0.
    2. Show that r must be a double root of the equation
      (x – a) (x – b) (x – c) (x – d) – p2 = 0.
  3. If sin x + sin y + sin z = 0 and cos x + cos y + cos z = 0, prove the following statements.
    1. cos (x – y) = –1/2
    2. cos (q – x) + cos (q – y) + cos (q – z) = 0, for any angle q.
    3. sin2 x + sin2 y + sin2 z = 3/2
    1. Construct an infinite collection { Ai } of infinite subsets of the set of natural numbers such that | Ai ∩ Aj | = 0 for i ≠ j.
    2. (b) Construct an infinite collection { Bi } of infinite subsets of the set of natural numbers such that | Bi ∩ Bj | gives a distinct integer for every pair of i and j, i ≠ j.
  5. Consider the equation x5 + y5 = z5.
    1. Show that the equation has a solution where x, y, and z are positive integers.
    2. Show that the equation has infinitely many solutions where x, y, and z are positive integers.



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