Difference between revisions of "2005 USAMO Problems"
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− | + | == Problem 1 == | |
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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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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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+ | == 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]] | |
− | + | = Day 2 = | |
− | + | == Problem 4 == | |
− | + | {{solution}} | |
[[2005 USAMO Problems/Problem 4 | Solution]] | [[2005 USAMO Problems/Problem 4 | Solution]] | ||
− | + | == Problem 5 == | |
+ | {{solution}} | ||
[[2005 USAMO Problems/Problem 5 | Solution]] | [[2005 USAMO Problems/Problem 5 | Solution]] | ||
− | + | == Problem 6 == | |
+ | {{solution}} | ||
[[2005 USAMO Problems/Problem 6 | Solution]] | [[2005 USAMO Problems/Problem 6 | Solution]] | ||
− | + | = Resources = | |
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* [[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] | ||
+ | {{USAMO newbox|year=2006|before=[[2005 USAMO]]|after=2007 USAMO}} |
Revision as of 13:03, 3 May 2008
Contents
Day 1
Problem 1
Determine all composite positive integers for which it is possible to arrange all divisors of that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.
Problem 2
Prove that the system has no solutions in integers , , and .
Problem 3
Let be an acute-angled triangle, and let and be two points on side . Construct point in such a way that convex quadrilateral is cyclic, , and and lie on opposite sides of line . Construct point in such a way that convex quadrilateral is cyclic, , and and lie on opposite sides of line . Prove that points , and lie on a circle.
Day 2
Problem 4
This problem needs a solution. If you have a solution for it, please help us out by adding it.
Problem 5
This problem needs a solution. If you have a solution for it, please help us out by adding it.
Problem 6
This problem needs a solution. If you have a solution for it, please help us out by adding it.
Resources
- USAMO Problems and Solutions
- 2005 USAMO Problems
- 2005 USAMO Solutions
- USAMO Problems on the Resources page
2006 USAMO (Problems • Resources) | ||
Preceded by 2005 USAMO |
Followed by 2007 USAMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |