Difference between revisions of "2005 USAMO Problems"

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Problems from the [[2005 USAMO | 2005]] [[USAMO]].
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= Day 1 =
 
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== Problem 1 ==
== Day 1 ==
 
 
 
=== Problem 1 ===
 
 
 
 
Determine all composite positive integers <math>n</math> for which it is possible to arrange all divisors of <math>n</math> that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.
 
Determine all composite positive integers <math>n</math> for which it is possible to arrange all divisors of <math>n</math> that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.
  
 
[[2005 USAMO Problems/Problem 1 | Solution]]
 
[[2005 USAMO Problems/Problem 1 | Solution]]
  
=== Problem 2 ===
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== Problem 2 ==
 
 
 
Prove that the
 
Prove that the
 
system
 
system
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has no solutions in integers <math>x</math>, <math>y</math>, and <math>z</math>.
 
has no solutions in integers <math>x</math>, <math>y</math>, and <math>z</math>.
  
* [[2005 USAMO Problems/Problem 2 | Solution]]
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[[2005 USAMO Problems/Problem 2 | Solution]]
 
 
=== Problem 3 ===
 
  
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== Problem 3 ==
 
Let <math>ABC</math> be an acute-angled triangle, and let <math>P</math> and <math>Q</math> be two points on side <math>BC</math>. Construct point <math>C_1 </math> in such a way that convex quadrilateral <math>APBC_1</math> is cyclic, <math>QC_1 \parallel CA</math>, and <math>C_1</math> and <math>Q</math> lie on opposite sides of line <math>AB</math>. Construct point <math>B_1</math> in such a way that convex quadrilateral <math>APCB_1</math> is cyclic, <math>QB_1 \parallel BA </math>, and <math>B_1 </math> and <math>Q </math>  lie on opposite sides of line <math>AC</math>.  Prove that points <math>B_1, C_1,P</math>, and <math>Q</math> lie on a circle.
 
Let <math>ABC</math> be an acute-angled triangle, and let <math>P</math> and <math>Q</math> be two points on side <math>BC</math>. Construct point <math>C_1 </math> in such a way that convex quadrilateral <math>APBC_1</math> is cyclic, <math>QC_1 \parallel CA</math>, and <math>C_1</math> and <math>Q</math> lie on opposite sides of line <math>AB</math>. Construct point <math>B_1</math> in such a way that convex quadrilateral <math>APCB_1</math> is cyclic, <math>QB_1 \parallel BA </math>, and <math>B_1 </math> and <math>Q </math>  lie on opposite sides of line <math>AC</math>.  Prove that points <math>B_1, C_1,P</math>, and <math>Q</math> lie on a circle.
  
* [[2005 USAMO Problems/Problem 3 | Solution]]
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[[2005 USAMO Problems/Problem 3 | Solution]]
  
== Day 2 ==
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= Day 2 =
 
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== Problem 4 ==
=== Problem 4 ===
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{{solution}}
  
 
[[2005 USAMO Problems/Problem 4 | Solution]]
 
[[2005 USAMO Problems/Problem 4 | Solution]]
  
=== Problem 5 ===
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== Problem 5 ==
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{{solution}}
  
 
[[2005 USAMO Problems/Problem 5 | Solution]]
 
[[2005 USAMO Problems/Problem 5 | Solution]]
  
=== Problem 6 ===
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== Problem 6 ==
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{{solution}}
  
 
[[2005 USAMO Problems/Problem 6 | Solution]]
 
[[2005 USAMO Problems/Problem 6 | Solution]]
  
== Resources ==
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= Resources =
 
 
 
* [[USAMO Problems and Solutions]]
 
* [[USAMO Problems and Solutions]]
 
* [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2006-ua/2006usamoQ.pdf 2005 USAMO Problems]
 
* [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2006-ua/2006usamoQ.pdf 2005 USAMO Problems]
 
* [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2006-ua/2006usamoS.pdf 2005 USAMO Solutions]
 
* [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2006-ua/2006usamoS.pdf 2005 USAMO Solutions]
 
* [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=2005 USAMO Problems on the Resources page]
 
* [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=2005 USAMO Problems on the Resources page]
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{{USAMO newbox|year=2006|before=[[2005 USAMO]]|after=2007 USAMO}}

Revision as of 13:03, 3 May 2008

Day 1

Problem 1

Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.

Solution

Problem 2

Prove that the system \begin{align*} x^6+x^3+x^3y+y & = 147^{157} \\ x^3+x^3y+y^2+y+z^9 & = 157^{147} \end{align*} has no solutions in integers $x$, $y$, and $z$.

Solution

Problem 3

Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on side $BC$. Construct point $C_1$ in such a way that convex quadrilateral $APBC_1$ is cyclic, $QC_1 \parallel CA$, and $C_1$ and $Q$ lie on opposite sides of line $AB$. Construct point $B_1$ in such a way that convex quadrilateral $APCB_1$ is cyclic, $QB_1 \parallel BA$, and $B_1$ and $Q$ lie on opposite sides of line $AC$. Prove that points $B_1, C_1,P$, and $Q$ lie on a circle.

Solution

Day 2

Problem 4

This problem needs a solution. If you have a solution for it, please help us out by adding it.

Solution

Problem 5

This problem needs a solution. If you have a solution for it, please help us out by adding it.

Solution

Problem 6

This problem needs a solution. If you have a solution for it, please help us out by adding it.

Solution

Resources

2006 USAMO (ProblemsResources)
Preceded by
2005 USAMO
Followed by
2007 USAMO
1 2 3 4 5 6
All USAMO Problems and Solutions
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