Art of Problem Solving
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2005 USAMO Problems: Difference between revisions

Newer edit →
No edit summary
 
m cosmetics
Line 5: Line 5:
=== Problem 1 ===
=== Problem 1 ===


Determine all composite positive integers <math> \displaystyle 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]]
Line 13: Line 13:
Prove that the
Prove that the
system
system
<center>
<cmath>
<math>
\begin{align*}
\begin{matrix} \qquad x^6+x^3+x^3y+y & = 147^{157} \\[.1in]
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{matrix}
\end{align*}
</math>
</cmath>
</center>
has no solutions in integers <math>x</math>, <math>y</math>, and <math>z</math>.
has no solutions in integers <math> \displaystyle x </math>, <math> \displaystyle y </math>, and <math> \displaystyle z </math>.
 
* [[2005 USAMO Problems/Problem 2 | Solution]]


=== Problem 3 ===
=== Problem 3 ===


Let <math> \displaystyle ABC </math> be an acute-angled triangle, and let <math> \displaystyle P </math> and <math> \displaystyle Q </math> be two points on side <math> \displaystyle BC </math>. Construct point <math> \displaystyle C_1 </math> in such a way that convex quadrilateral <math> \displaystyle APBC_1 </math> is cyclic, <math> \displaystyle QC_1 \mid\mid CA </math>, and <math> \displaystyle C_1 </math> and <math> \displaystyle Q </math> lie on opposite sides of line <math> \displaystyle AB </math>. Construct point <math> \displaystyle B_1 </math> in such a way that convex quadrilateral <math> \displaystyle APCB_1 </math> is cyclic, <math> \displaystyle QB_1 \mid\mid BA </math>, and <math> \displaystyle B_1 </math> and <math> \displaystyle Q </math>  lie on opposite sides of line <math> \displaystyle AC </math>.  Prove that points <math> \displaystyle B_1, C_1,P </math>, and <math> \displaystyle 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 ==
== Day 2 ==

Revision as of 20:53, 11 April 2008

Problems from the 2005 USAMO.

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 $z$.

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.

Day 2

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

Resources

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2005_USAMO_Problems&oldid=24683"