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

Difference between revisions of "2004 USAMO Problems"

m
Line 5: Line 5:
 
=== Problem 1 ===
 
=== Problem 1 ===
  
−
Let <math> \displaystyle ABCD </math> be a [[quadrilateral]] circumscribed about a circle, whose interior and exterior angles are at least <math> \displaystyle 60 ^{\circ} </math>.  Prove that
+
Let <math>ABCD </math> be a [[quadrilateral]] circumscribed about a circle, whose interior and exterior angles are at least <math>60 ^{\circ} </math>.  Prove that
 
<center>
 
<center>
 
<math>
 
<math>
Line 18: Line 18:
 
=== Problem 2 ===
 
=== Problem 2 ===
  
−
Suppose <math> a_1, \ldots, a_n </math> are integers whose greatest common divisor is 1.  Let <math> \displaystyle S </math> be a sequence of integers with the following properties:
+
Suppose <math> a_1, \ldots, a_n </math> are integers whose greatest common divisor is 1.  Let <math>S </math> be a sequence of integers with the following properties:
  
 
(a) For <math> i = 1, \ldots , n </math>, <math> a_i \in S </math>. <br>
 
(a) For <math> i = 1, \ldots , n </math>, <math> a_i \in S </math>. <br>
Line 24: Line 24:
 
(c) For any integers <math> x, y \in S </math>, if <math> x+y \in S </math>, then <math> x-y \in S </math>.
 
(c) For any integers <math> x, y \in S </math>, if <math> x+y \in S </math>, then <math> x-y \in S </math>.
  
−
Prove that <math> \displaystyle S </math> must be the set of all integers.
+
Prove that <math>S </math> must be the set of all integers.
  
 
[[2004 USAMO Problems/Problem 2 | Solution]]
 
[[2004 USAMO Problems/Problem 2 | Solution]]
Line 30: Line 30:
 
=== Problem 3 ===
 
=== Problem 3 ===
  
−
For what values of <math> \displaystyle k > 0 </math> is it possible to dissect a <math> 1 \times k </math> rectangle into two similar, but incongruent, polygons?
+
For what values of <math>k > 0 </math> is it possible to dissect a <math> 1 \times k </math> rectangle into two similar, but incongruent, polygons?
  
 
[[2004 USAMO Problems/Problem 3 | Solution]]
 
[[2004 USAMO Problems/Problem 3 | Solution]]
Line 44: Line 44:
 
=== Problem 5 ===
 
=== Problem 5 ===
  
−
Let <math> \displaystyle a </math>, <math> \displaystyle b </math>, and <math> \displaystyle c </math> be positive real numbers.  Prove that
+
Let <math>a </math>, <math>b </math>, and <math>c </math> be positive real numbers.  Prove that
 
<center>
 
<center>
 
<math>
 
<math>
Line 55: Line 55:
 
=== Problem 6 ===
 
=== Problem 6 ===
  
−
A circle <math> \displaystyle \omega </math> is inscribed in a quadrilateral <math> \displaystyle ABCD </math>.  Let <math> \displaystyle I </math> be the center of <math> \displaystyle \omega </math>.  Suppose that
+
A circle <math>\omega </math> is inscribed in a quadrilateral <math>ABCD </math>.  Let <math>I </math> be the center of <math>\omega </math>.  Suppose that
 
<center>
 
<center>
 
<math>
 
<math>
−
\displaystyle (AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2
+
(AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2
 
</math>.
 
</math>.
 
</center>
 
</center>
−
Prove that <math> \displaystyle ABCD </math> is an [[isosceles trapezoid]].
+
Prove that <math>ABCD </math> is an [[isosceles trapezoid]].
  
 
[[2004 USAMO Problems/Problem 6 | Solution]]
 
[[2004 USAMO Problems/Problem 6 | Solution]]
Line 71: Line 71:
 
* [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2004-ua/04USAMO_solutions.pdf 2004 USAMO Solutions]
 
* [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2004-ua/04USAMO_solutions.pdf 2004 USAMO Solutions]
 
* [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=2004 2004 USAMO Problems on the Resources page]
 
* [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=2004 2004 USAMO Problems on the Resources page]
 +
{{MAA Notice}}

Revision as of 12:39, 4 July 2013

Problems from the 2004 USAMO.

Day 1

Problem 1

Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least $60 ^{\circ}$. Prove that

$\frac{1}{3} | AB^3 - AD^3 | \le | BC^3 - CD^3 | \le 3 |AB^3 - AD^3 |$

When does equality hold?

Solution

Problem 2

Suppose $a_1, \ldots, a_n$ are integers whose greatest common divisor is 1. Let $S$ be a sequence of integers with the following properties:

(a) For $i = 1, \ldots , n$, $a_i \in S$.
(b) For $i, j = 1, \ldots, n$ (not necessarily distinct), $a_i - a_j \in S$.
(c) For any integers $x, y \in S$, if $x+y \in S$, then $x-y \in S$.

Prove that $S$ must be the set of all integers.

Solution

Problem 3

For what values of $k > 0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but incongruent, polygons?

Solution

Day 2

Problem 4

Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing on the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest number is colored black. Alice wins if she can then draw a line from the top of the grid to the bottom of the grid that stays in black squares, and Bob wins if she can't. (If two squares share a vertex, Alice can draw a line from on to the other that stays in those two squares. Find, with proof, a winning strategy for one of the players.

Solution

Problem 5

Let $a$, $b$, and $c$ be positive real numbers. Prove that

$(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \ge (a+b+c)^3$.

Solution

Problem 6

A circle $\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\omega$. Suppose that

$(AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2$.

Prove that $ABCD$ is an isosceles trapezoid.

Solution

Resources

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2004_USAMO_Problems&oldid=54779"