The Eighth Canadian Mathematical Olympiads
1976年第八届加拿大数学奥林匹克 

 Given four unequal weights in geometric progression, show how to find the heaviest weight using a balance twice.
 The real sequence x_{0}, x_{1}, x_{2}, ... is defined by x_{0} = 1, x_{1} = 2, n(n+1) x_{n+1} = n(n1) x_{n}  (n2) x_{n1}. Find x_{0}/x_{1} + x_{1}/x_{2} + ... + x_{50}/x_{51}.
 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.
 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.
 Show that a positive integer is a sum of two or more consecutive positive integers iff it is not a power of 2.
 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.
 p(x, y) is a symmetric polynomial with the factor (x  y). Show that (x  y)^{2} is a factor.
 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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