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Difference between revisions of "1980 Canadian MO"

(Created page with "== Problem 1 == If <math>a679b</math> is the decimal expansion of a number in base <math>10</math>, such that it is divisible by <math>72</math>, determine <math> a,b</math>. ...")
 
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== Problem 1 ==
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'''1980 Canadian MO''' problems and solutions.  The first link contains the full set of test problems.  The rest contain each individual problem and its solution.
  
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If <math>a679b</math> is the decimal expansion of a number in base <math>10</math>, such that it is divisible by <math>72</math>, determine <math> a,b</math>.
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* [[1980 Canadian MO Problems]]
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* [[1980 Canadian MO Problems/Problem 1]]
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* [[1980 Canadian MO Problems/Problem 2]]
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* [[1980 Canadian MO Problems/Problem 3]]
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* [[1980 Canadian MO Problems/Problem 4]]
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* [[1980 Canadian MO Problems/Problem 5]]
  
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[[1980 Canadian MO Problems/Problem 1 | Solution]]
 
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== Problem 2 ==
 
  
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The numbers from <math>1</math> to <math>50</math> are printed on cards. The cards are shuffled and then laid out face up in <math>5</math> rows of <math>10</math> cards each. The cards in each row are rearranged to make them increase from left to right. The cards in each column are then rearranged to make them increase from top to bottom. In the final arrangement, do the cards in the rows still increase from left to right?
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== See also ==
  
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[[1980 Canadian MO Problems/Problem 2 | Solution]]
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* [[Mathematics competitions]]
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== Problem 3 ==
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* [[Mathematics competition resources]]
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* [[Math books]]
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Among all triangles having
 
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(i) a fixed angle <math>A</math> and
 
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(ii) an inscribed circle of fixed radius <math>r</math>, determine which triangle has the least minimum perimeter.
 
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[[180 Canadian MO Problems/Problem 3 | Solution]]
 
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== Problem 4 ==
 
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Let <math>ABC</math> be an equilateral triangle, and <math>P</math> be an arbitrary point within the triangle. Perpendiculars <math>PD,PE,PF</math> are drawn to the three sides of the triangle. Show that, no matter where <math>P</math> is chosen, <math>\frac{PD+PE+PF}{AB+BC+CA}=\frac{1}{2\sqrt{3}}</math>.
 
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[[1980 Canadian MO Problems/Problem 4 | Solution]]
 
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== Problem 5 ==
 
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A gambling student tosses a fair coin. She gains <math>1</math> point for each head that turns up, and gains <math>2</math> points for each tail that turns up. Prove that the probability of the student scoring exactly <math>n</math> points is <math>\frac{1}{3}\cdot\left(2+\left(-\frac{1}{2}\right)^{n}\right)</math>.
 
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[[1980 Canadian MO Problems/Problem 5 | Solution]]
 
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A parallelepiped has the property that all cross sections, which are parallel to any fixed face <math>F</math>, have the same perimeter as <math>F</math>. Determine whether or not any other polyhedron has this property.
 
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[[1980 Canadian MO Problems/Problem 10 | Solution]]
 
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== Resources ==
 
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* [[1980 Canadian MO]]
 
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* [[Canadian Mathematical Olympiad]]
 
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* [[Canadian MO Problems and Solutions]]
 

Latest revision as of 21:38, 14 December 2011

1980 Canadian MO problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.


See also

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