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The Eighteenth Canadian Mathematical Olympiads
1986年第十八届加拿大数学奥林匹克
  1. The triangle ABC has angle B = 90o. The point D is taken on the ray AC, the other side of C from A, such that CD = AB. ∠CBD = 30o. Find AC/CD.
  2. Three competitors A, B, C compete in a number of sporting events. In each event a points is awarded for a win, b points for second place and c points for third place. There are no ties. The final score was A 22, B 9, C 9. B won the 100 meters. How many events were there and who came second in the high jump ?
  3. A chord AB of constant length slides around the curved part of a semicircle. M is the midpoint of AB, and C is the foot of the perpendicular from A onto the diameter. Show that angle ACM does not change.
  4. Show that (1 + 2 + ... + n) divides (1k + 2k + ... + nk) for k odd.
  5. The integer sequence a1, a2, a3, ... is defined by a1 = 39, a2 = 45, an+2 = an+12 - an. Show that infinitely many terms of the sequence are divisible by 1986.
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