Difference between revisions of "1971 Canadian MO Problems"

Revision as of 21:49, 13 December 2011

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.

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

Resources

Revision as of 21:47, 13 December 2011 (view source) Airplanes1 (talk \| contribs) (→‎Problem 6) ← Older edit		Revision as of 21:49, 13 December 2011 (view source) Airplanes1 (talk \| contribs) (→‎Problem 5) Newer edit →
Line 29:		Line 29:
	== Problem 5 ==		== Problem 5 ==

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

Page

Toolbox

Search

Difference between revisions of "1971 Canadian MO Problems"

Revision as of 21:49, 13 December 2011

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Resources