45th MMPC Part II Problems

The top 1171 students had 100 minutes to solve these five problems and compete for scholarships and recognition.
    1. Show that for every positive integer m > 1, there are positive integers x and y such that
      x2 – y 2 = m 3.
    2. Find all pairs of positive integers (x,y) such that
      x6 = y 2 + 127.
    1. Let P(x) be a polynomial with integer coefficients. Suppose that P(0) is an odd integer and that P(1) is also an odd integer. Show that if c is an integer then P(c) is not equal to 0.
    2. Let P(x) be a polynomial with integer coefficients. Suppose that P(1,000) = 1,000 and P(2,000) = 2,000. Explain why P(3,000) cannot be equal to 1,000.
  3. Triangle ABC is created from points A(0,0), B(1,0) and C(1/2,2). Let q, r, and s be numbers such that 0 < q < 1/2 < s < 1, and q < r < s. Let D be the point on AC which has x-coordinate q, E be the point on AB which has x-coordinate r, and F be the point on BC that has x-coordinate s.
    1. Find the area of triangle DEF in terms of q, r, and s.
    2. If r = 1/2, prove that at least one of the triangles ADE, CDF, or BEF has an area of at least 1/4.
  4. In the Gregorian calendar:
    • years not divisible by 4 are common years,
    • years divisible by 4 but not by 100 are leap years,
    • years divisible by 100 but not by 400 are common years,
    • years divisible by 400 are leap years,
    • a leap year contains 366 days; a common year 365 days.
    From the information above:
    1. Find the number of common years and leap years in 400 consecutive Gregorian years. Show that 400 consecutive Gregorian years consists of an integral number of weeks.
    2. Prove that the probability that Christmas falls on a Wednesday is not equal to 1/7.
  5. Each of the first 13 letters of the alphabet is written on the back of a card and the 13 cards are placed in a row in the order
    A, B, C, D, E, F, G, H, I, J, K, L, M.
    The cards are then turned over so that the letters are face down. The cards are rearranged and again placed in a row, but of course they may be in a different order. They are rearranged and placed in a row a second time and both rearrangements were performed exactly the same way. When the cards are turned over the letters are in the order
    B, M, A, H, G, C, F, E, D, L, I, K, J
    What was the order of the letters after the cards were rearranged the first time?



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