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The Eighth Canadian Mathematical Olympiads
1976年第八届加拿大数学奥林匹克
  1. Given four unequal weights in geometric progression, show how to find the heaviest weight using a balance twice.
  2. The real sequence x0, x1, x2, ... is defined by x0 = 1, x1 = 2, n(n+1) xn+1 = n(n-1) xn - (n-2) xn-1. Find x0/x1 + x1/x2 + ... + x50/x51.
  3. n+2 students played a tournament. Each pair played each other once. A player scored 1 for a win, 1/2 for a draw and nil for a loss. Two students scored a total of 8 and the other players all had equal total scores. Find n.
  4. C lies on the segment AB. P is a variable point on the circle with diameter AB. Q lies on the line CP on the opposite side of C to P such that PC/CQ = AC/CB. Find the locus of Q.
  5. Show that a positive integer is a sum of two or more consecutive positive integers iff it is not a power of 2.
  6. The four points A, B, C, D in space are such that the angles ABC, BCD, CDA, DAB are all right angles. Show that the points are coplanar.
  7. p(x, y) is a symmetric polynomial with the factor (x - y). Show that (x - y)2 is a factor.
  8. A graph has 9 points and 36 edges. Each edge is colored red or blue. Every triangle has a red edge. Show that there are four points with all edges between them red.
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