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== | ||
− | 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>. | + | 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>. |
* Prove that <math>(c/a + c/b)^2 > 8</math>. | * 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>. | * 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>. | ||
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[[2005 Canadian MO Problems/Problem 2 | Solution]] | [[2005 Canadian MO Problems/Problem 2 | Solution]] | ||
==Problem 3== | ==Problem 3== | ||
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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>. | ||
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[[2005 Canadian MO Problems/Problem 4 | Solution]] | [[2005 Canadian MO Problems/Problem 4 | Solution]] | ||
==Problem 5== | ==Problem 5== | ||
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* 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. | ||
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[[2005 Canadian MO Problems/Problem 5 | Solution]] | [[2005 Canadian MO Problems/Problem 5 | Solution]] | ||
== Resources == | == Resources == | ||
[[2005 Canadian MO]] | [[2005 Canadian MO]] |
Revision as of 18:51, 28 July 2006
Problem 1
Consider an equilateral triangle of side length , which is divided into unit triangles, as shown. Let
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
. Determine the value of
.
Problem 2
Let be a Pythagorean triple, i.e., a triplet of positive integers with
.
- Prove that
.
- Prove that there does not exist any integer
for which we can find a Pythagorean triple
satisfying
.
Problem 3
Let be a set of
points in the interior of a circle.
- Show that there are three distinct points
and three distinct points
on the circle such that
is (strictly) closer to
than any other point in
,
is closer to
than any other point in
and
is closer to
than any other point in
.
- Show that for no value of
can four such points in
(and corresponding points on the circle) be guaranteed.
Problem 4
Let be a triangle with circumradius
, perimeter
and area
. Determine the maximum value of
.
Problem 5
Let's say that an ordered triple of positive integers is
-\emph{powerful} if
,
, and
is divisible by
. For example,
is 5-powerful.
- Determine all ordered triples (if any) which are
-powerful for all
.
- Determine all ordered triples (if any) which are 2004-powerful and 2005-powerful, but not 2007-powerful.