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Difference between revisions of "2005 Canadian MO Problems"

 
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[[Image:CanMO_2005_1.png]]
 
[[Image:CanMO_2005_1.png]]
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[[2005 Canadian MO Problems/Problem 1 | Solution]]
 
[[2005 Canadian MO Problems/Problem 1 | Solution]]
 
==Problem 2==
 
==Problem 2==
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Let <math>(a,b,c)</math> be a Pythagorean triple, ''i.e.'', a triplet of positive integers with <math>a^2+b^2=c^2</math>.
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Let <math>(a,b,c)</math> be a Pythagorean triple, ''i.e.'', a triplet of positive integers with <math>{a}^2+{b}^2={c}^2</math>.
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 +
* Prove that <math>\left(\frac{c}{a}+\frac{c}{b}\right)^2>8</math>.
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* Prove that there are no integer <math>n</math> and Pythagorean triple <math>(a,b,c)</math> satisfying <math>\left(\frac{c}{a}+\frac{c}{b}\right)^2=n</math>.
  
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* Prove that <math>(c/a + c/b)^2 > 8</math>.
 
−
* Prove that there does not exist any integer <math>n</math> for which we can find a Pythagorean triple <math>(a,b,c)</math> satisfying <math>(c/a + c/b)^2 = n</math>.
 
 
[[2005 Canadian MO Problems/Problem 2 | Solution]]
 
[[2005 Canadian MO Problems/Problem 2 | Solution]]
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==Problem 3==
 
==Problem 3==
 
Let <math>S</math> be a set of <math>n\ge 3</math> points in the interior of a circle.  
 
Let <math>S</math> be a set of <math>n\ge 3</math> points in the interior of a circle.  
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==Problem 4==
 
==Problem 4==
 
Let <math>ABC</math> be a triangle with circumradius <math>R</math>, perimeter <math>P</math> and area <math>K</math>. Determine the maximum value of <math>KP/R^3</math>.
 
Let <math>ABC</math> be a triangle with circumradius <math>R</math>, perimeter <math>P</math> and area <math>K</math>. Determine the maximum value of <math>KP/R^3</math>.
 +
 
[[2005 Canadian MO Problems/Problem 4 | Solution]]
 
[[2005 Canadian MO Problems/Problem 4 | Solution]]
 
==Problem 5==
 
==Problem 5==
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Let's say that an ordered triple of positive integers <math>(a,b,c)</math> is <math>n</math>-\emph{powerful} if <math>a \le b \le c</math>, <math>\gcd(a,b,c) = 1</math>, and  <math>a^n + b^n + c^n</math> is divisible by <math>a+b+c</math>. For example, <math>(1,2,2)</math> is 5-powerful.
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Let's say that an ordered triple of positive integers <math>(a,b,c)</math> is <math>n</math>-''powerful'' if <math>a \le b \le c</math>, <math>\gcd(a,b,c) = 1</math>, and  <math>a^n + b^n + c^n</math> is divisible by <math>a+b+c</math>. For example, <math>(1,2,2)</math> is 5-powerful.
  
 
* Determine all ordered triples (if any) which are <math>n</math>-powerful for all <math>n \ge 1</math>.
 
* Determine all ordered triples (if any) which are <math>n</math>-powerful for all <math>n \ge 1</math>.
 
* Determine all ordered triples (if any) which are 2004-powerful and 2005-powerful, but not 2007-powerful.
 
* Determine all ordered triples (if any) which are 2004-powerful and 2005-powerful, but not 2007-powerful.
 +
 
[[2005 Canadian MO Problems/Problem 5 | Solution]]
 
[[2005 Canadian MO Problems/Problem 5 | Solution]]
 +
 
== Resources ==
 
== Resources ==
 
[[2005 Canadian MO]]
 
[[2005 Canadian MO]]

Latest revision as of 11:45, 26 December 2018

Problem 1

Consider an equilateral triangle of side length $n$, which is divided into unit triangles, as shown. Let $f(n)$ be the number of paths from the triangle in the top row to the middle triangle in the bottom row, such that adjacent triangles in our path share a common edge and the path never travels up (from a lower row to a higher row) or revisits a triangle. An example of one such path is illustrated below for $n=5$. Determine the value of $f(2005)$.

CanMO 2005 1.png

Solution

Problem 2

Let $(a,b,c)$ be a Pythagorean triple, i.e., a triplet of positive integers with ${a}^2+{b}^2={c}^2$.

  • Prove that $\left(\frac{c}{a}+\frac{c}{b}\right)^2>8$.
  • Prove that there are no integer $n$ and Pythagorean triple $(a,b,c)$ satisfying $\left(\frac{c}{a}+\frac{c}{b}\right)^2=n$.

Solution

Problem 3

Let $S$ be a set of $n\ge 3$ points in the interior of a circle.

  • Show that there are three distinct points $a,b,c\in S$ and three distinct points $A,B,C$ on the circle such that $a$ is (strictly) closer to $A$ than any other point in $S$, $b$ is closer to $B$ than any other point in $S$ and $c$ is closer to $C$ than any other point in $S$.
  • Show that for no value of $n$ can four such points in $S$ (and corresponding points on the circle) be guaranteed.

Solution

Problem 4

Let $ABC$ be a triangle with circumradius $R$, perimeter $P$ and area $K$. Determine the maximum value of $KP/R^3$.

Solution

Problem 5

Let's say that an ordered triple of positive integers $(a,b,c)$ is $n$-powerful if $a \le b \le c$, $\gcd(a,b,c) = 1$, and $a^n + b^n + c^n$ is divisible by $a+b+c$. For example, $(1,2,2)$ is 5-powerful.

  • Determine all ordered triples (if any) which are $n$-powerful for all $n \ge 1$.
  • Determine all ordered triples (if any) which are 2004-powerful and 2005-powerful, but not 2007-powerful.

Solution

Resources

2005 Canadian MO

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