Difference between revisions of "1969 Canadian MO Problems"
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== Problem 1 == | == Problem 1 == | ||
− | Show that if <math> | + | Show that if <math>a_1/b_1=a_2/b_2=a_3/b_3</math> and <math>p_1,p_2,p_3</math> are not all zero, then <math>\left(\frac{a_1}{b_1} \right)^n=\frac{p_1a_1^n+p_2a_2^n+p_3a_3^n}{p_1b_1^n+p_2b_2^n+p_3b_3^n}</math> for every positive integer <math>n.</math> |
+ | [[1969 Canadian MO Problems/Problem 1 | Solution]] | ||
== Problem 2 == | == Problem 2 == | ||
− | Determine which of the two numbers <math> | + | Determine which of the two numbers <math>\sqrt{c+1}-\sqrt{c}</math>, <math>\sqrt{c}-\sqrt{c-1}</math> is greater for any <math>c\ge 1</math>. |
+ | [[1969 Canadian MO Problems/Problem 2 | Solution]] | ||
== Problem 3 == | == Problem 3 == | ||
− | Let <math> | + | Let <math>c</math> be the length of the [[hypotenuse]] of a [[right triangle]] whose two other sides have lengths <math>a</math> and <math>b</math>. Prove that <math>a+b\le c\sqrt{2}</math>. When does the equality hold? |
+ | [[1969 Canadian MO Problems/Problem 3 | Solution]] | ||
== Problem 4 == | == Problem 4 == | ||
− | Let <math> | + | 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>. |
+ | [[1969 Canadian MO Problems/Problem 4 | Solution]] | ||
== Problem 5 == | == Problem 5 == | ||
− | Let <math> | + | Let <math>ABC</math> be a triangle with sides of length <math>a</math>, <math>b</math> and <math>c</math>. Let the bisector of the <math>\angle C</math> cut <math>AB</math> at <math>D</math>. Prove that the length of <math>CD</math> is <math>\frac{2ab\cos \frac{C}{2}}{a+b}.</math> |
+ | [[1969 Canadian MO Problems/Problem 5 | Solution]] | ||
== Problem 6 == | == Problem 6 == | ||
− | Find the sum of <math> | + | Find the sum of <math>1\cdot 1!+2\cdot 2!+3\cdot 3!+\cdots+(n-1)(n-1)!+n\cdot n!</math>, where <math> n!=n(n-1)(n-2)\cdots2\cdot1</math>. |
+ | [[1969 Canadian MO Problems/Problem 6 | Solution]] | ||
== Problem 7 == | == Problem 7 == | ||
− | Show that there are no integers <math> | + | Show that there are no integers <math>a,b,c</math> for which <math>a^2+b^2-8c=6</math>. |
− | |||
+ | [[1969 Canadian MO Problems/Problem 7 | Solution]] | ||
== Problem 8 == | == Problem 8 == | ||
− | Let <math> | + | Let <math>f</math> be a function with the following properties: |
− | 1) <math> | + | 1) <math>f(n)</math> is defined for every positive integer <math>n</math>; |
− | 2) <math> | + | 2) <math>f(n)</math> is an integer; |
− | 3) <math> | + | 3) <math>f(2)=2</math>; |
− | 4) <math> | + | 4) <math>f(mn)=f(m)f(n)</math> for all <math>m</math> and <math>n</math>; |
− | 5) <math> | + | 5) <math>f(m)>f(n)</math> whenever <math>m>n</math>. |
− | Prove that <math> | + | Prove that <math>f(n)=n</math>. |
+ | [[[1969 Canadian MO Problems/Problem 8 | Solution]] | ||
== Problem 9 == | == Problem 9 == | ||
− | Show that for any quadrilateral inscribed in a circle of radius <math> | + | Show that for any quadrilateral inscribed in a circle of radius <math>1,</math> the length of the shortest side is less than or equal to <math>\sqrt{2}</math>. |
+ | [[1969 Canadian MO Problems/Problem 9 | Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
− | Let <math> | + | Let <math>ABC</math> be the right-angled isosceles triangle whose equal sides have length 1. <math>P</math> is a point on the [[hypotenuse]], and the feet of the [[perpendicular]]s from <math>P</math> to the other sides are <math>Q</math> and <math>R</math>. Consider the areas of the triangles <math>APQ</math> and <math>PBR</math>, and the area of the [[rectangle]] <math>QCRP</math>. Prove that regardless of how <math>P</math> is chosen, the largest of these three areas is at least <math>2/9</math>. |
+ | [[1969 Canadian MO Problems/Problem 10 | Solution]] | ||
== Resources == | == Resources == | ||
Revision as of 11:46, 8 October 2007
Contents
Problem 1
Show that if and are not all zero, then for every positive integer
Problem 2
Determine which of the two numbers , is greater for any .
Problem 3
Let be the length of the hypotenuse of a right triangle whose two other sides have lengths and . Prove that . When does the equality hold?
Problem 4
Let be an equilateral triangle, and be an arbitrary point within the triangle. Perpendiculars are drawn to the three sides of the triangle. Show that, no matter where is chosen, .
Problem 5
Let be a triangle with sides of length , and . Let the bisector of the cut at . Prove that the length of is
Problem 6
Find the sum of , where .
Problem 7
Show that there are no integers for which .
Problem 8
Let be a function with the following properties:
1) is defined for every positive integer ;
2) is an integer;
3) ;
4) for all and ;
5) whenever .
Prove that .
[[[1969 Canadian MO Problems/Problem 8 | Solution]]
Problem 9
Show that for any quadrilateral inscribed in a circle of radius the length of the shortest side is less than or equal to .
Problem 10
Let be the right-angled isosceles triangle whose equal sides have length 1. is a point on the hypotenuse, and the feet of the perpendiculars from to the other sides are and . Consider the areas of the triangles and , and the area of the rectangle . Prove that regardless of how is chosen, the largest of these three areas is at least .