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Difference between revisions of "1971 Canadian MO Problems"

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Show that, for all integers <math>n</math>, <math>n^2+2n+12</math> is not a multiple of 121.  
 
Show that, for all integers <math>n</math>, <math>n^2+2n+12</math> is not a multiple of 121.  
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[[1971 Canadian MO Problems/Problem 6 | Solution]]
 
[[1971 Canadian MO Problems/Problem 6 | Solution]]
  

Revision as of 21:49, 13 December 2011

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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

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

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