Difference between revisions of "2005 USAMO Problems"
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=== Problem 1 === | === Problem 1 === | ||
− | Determine all composite positive integers <math> | + | 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]] | ||
Line 13: | Line 13: | ||
Prove that the | Prove that the | ||
system | system | ||
− | < | + | <cmath> |
− | + | \begin{align*} | |
− | \begin{ | + | x^6+x^3+x^3y+y & = 147^{157} \ |
x^3+x^3y+y^2+y+z^9 & = 157^{147} | x^3+x^3y+y^2+y+z^9 & = 157^{147} | ||
− | \end{ | + | \end{align*} |
− | </ | + | </cmath> |
− | + | has no solutions in integers <math>x</math>, <math>y</math>, and <math>z</math>. | |
− | has no solutions in integers <math> | + | |
+ | * [[2005 USAMO Problems/Problem 2 | Solution]] | ||
=== Problem 3 === | === Problem 3 === | ||
− | Let <math> | + | 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 == | == Day 2 == |
Revision as of 20:53, 11 April 2008
Contents
[hide]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.