Difference between revisions of "1971 Canadian MO Problems"

(Problem 4)
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== Problem 2 ==
 
== Problem 2 ==
  
 
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Let <math>x</math> and <math>y</math> be positive real numbers such that <math>x+y=1</math>. Show that <math>\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)\ge 9</math>.
  
 
[[1971 Canadian MO Problems/Problem 2 | Solution]]
 
[[1971 Canadian MO Problems/Problem 2 | Solution]]
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<math>ABCD</math> is a quadrilateral with <math>AD=BC</math>. If <math>\angle ADC</math> is greater than <math>\angle BCD</math>, prove that <math>AC>BD</math>.
  
 
[[1971 Canadian MO Problems/Problem 3 | Solution]]
 
[[1971 Canadian MO Problems/Problem 3 | Solution]]

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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

CanadianMO 1971-1.jpg


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

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

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