1971 Canadian MO Problems

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Problem 1

$DEB$ is a chord of a circle such that $DE=3$ and $EB=5 .$ Let $O$ be the center of the circle. Join $OE$ and extend $OE$ to cut the circle at $C.$ Given $EC=1,$ find the radius of the circle

Solution

Problem 2

Solution

Problem 3

Solution

Problem 4

Solution

Problem 5

A quadrilateral has one vertex on each side of a square of side-length 1. Show that the lengths $a$ , $b$ , $c$ and $d$ of the sides of the quadrilateral satisfy the inequalities $2\le a^2+b^2+c^2+d^2\le 4.$