1971 Canadian MO Problems/Problem 1


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

CanadianMO 1971-1.jpg


First, extend $\displaystyle CO$ to meet the circle at $\displaystyle P.$ Let the radius be $\displaystyle r.$ Applying power of a point, $\displaystyle (EP)(CE)=(BE)(ED)$ and $\displaystyle 2r-1=15.$ Hence, $\displaystyle r=8.$

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