Difference between revisions of "2005 USAMO Problems"
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== Problem 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. | + | (''Zuming Feng'') 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 == | == Problem 2 == | ||
| − | Prove that the | + | (''Răzvan Gelca'') Prove that the |
system | system | ||
<cmath> | <cmath> | ||
| Line 19: | Line 19: | ||
== Problem 3 == | == 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. | + | (''Zuming Feng'') 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]] | [[2005 USAMO Problems/Problem 3 | Solution]] | ||
Revision as of 12:28, 3 May 2008
Contents
Day 1
Problem 1
(Zuming Feng) 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
(Răzvan Gelca) Prove that the
system
has no solutions in integers 
, 
, and 
.
Problem 3
(Zuming Feng) 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
| 2005 USAMO (Problems • Resources) | ||
| Preceded by 2004 USAMO |
Followed by 2006 USAMO | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAMO Problems and Solutions | ||
