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

Let $x$ and $y$ be positive real numbers such that $x+y=1$ . Show that $\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)\ge 9$ .

Solution

Problem 3

$ABCD$ is a quadrilateral with $AD=BC$ . If $\angle ADC$ is greater than $\angle BCD$ , prove that $AC>BD$ .

Solution

Problem 4

Determine all real numbers $a$ such that the two polynomials $x^2+ax+1$ and $x^2+x+a$ have at least one root in common.

Solution

Problem 5

Solution

Problem 6

Show that, for all integers $n$ , $n^2+2n+12$ is not a multiple of 121. Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

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1971 Canadian MO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Resources