1971 Canadian MO Problems

This is an empty template page which needs to be filled. You can help us out by finding the needed content and editing it in. Thanks.

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

Let $p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x+a_0$ , where the coefficients $a_i$ are integers. If $p(0)$ and $p(1)$ are both odd, show that $p(x)$ has no integral roots.