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The Sixteenth Canadian Mathematical Olympiads
1984年第十六届加拿大数学奥林匹克
  1. Show that the sum of 1984 consecutive positive integers cannot be a square.
  2. You have keyring with n identical keys. You wish to color code the keys so that you can distinguish them. What is the smallest number of colors you need?
    [For example, you could use three colors for eight keys: R R R R G B R R. Starting with the blue key and moving away from the green key uniquely distinguishes each of the red keys.]
  3. Show that there are infinitely many integers which have no zeros and which are divisible by the sum of their digits.
  4. An acute-angled triangle has unit area. Show that there is a point inside the triangle which is at least 2/(33/4) from any vertex.
  5. Given any seven real numbers show we can select two, x and y, such that 0 ≤ (x - y)/(1 + xy) ≤ 1/√3.
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