Difference between revisions of "2004 USAMO Problems"
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=== Problem 1 === | === Problem 1 === | ||
− | Let <math> | + | 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> | ||
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=== Problem 2 === | === Problem 2 === | ||
− | Suppose <math> a_1, \ldots, a_n </math> are integers whose greatest common divisor is 1. Let <math> | + | Suppose <math> a_1, \ldots, a_n </math> are integers whose greatest common divisor is 1. Let <math>S </math> be a set 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> | ||
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(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> | + | Prove that <math>S </math> must be the set of all integers. |
[[2004 USAMO Problems/Problem 2 | Solution]] | [[2004 USAMO Problems/Problem 2 | Solution]] | ||
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=== Problem 3 === | === Problem 3 === | ||
− | For what values of <math> | + | 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]] | ||
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=== Problem 5 === | === Problem 5 === | ||
− | Let <math> | + | Let <math>a </math>, <math>b </math>, and <math>c </math> be positive real numbers. Prove that |
<center> | <center> | ||
<math> | <math> | ||
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</math>. | </math>. | ||
</center> | </center> | ||
+ | |||
+ | [[2004 USAMO Problems/Problem 5 | Solution]] | ||
=== Problem 6 === | === Problem 6 === | ||
− | A circle <math> | + | 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> | ||
− | + | (AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2 | |
</math>. | </math>. | ||
</center> | </center> | ||
− | Prove that <math> | + | Prove that <math>ABCD </math> is an [[isosceles trapezoid]]. |
[[2004 USAMO Problems/Problem 6 | Solution]] | [[2004 USAMO Problems/Problem 6 | Solution]] | ||
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* [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] | ||
+ | {{USAMO newbox|year=2004|before=[[2003 USAMO]]|after=[[2005 USAMO]]}} | ||
+ | {{MAA Notice}} |
Latest revision as of 08:20, 14 May 2021
Contents
Day 1
Problem 1
Let be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least . Prove that
When does equality hold?
Problem 2
Suppose are integers whose greatest common divisor is 1. Let be a set of integers with the following properties:
(a) For , .
(b) For (not necessarily distinct), .
(c) For any integers , if , then .
Prove that must be the set of all integers.
Problem 3
For what values of is it possible to dissect a rectangle into two similar, but incongruent, polygons?
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.
Problem 5
Let , , and be positive real numbers. Prove that
.
Problem 6
A circle is inscribed in a quadrilateral . Let be the center of . Suppose that
.
Prove that is an isosceles trapezoid.
Resources
- USAMO Problems and Solutions
- 2004 USAMO Problems
- 2004 USAMO Solutions
- 2004 USAMO Problems on the Resources page
2004 USAMO (Problems • Resources) | ||
Preceded by 2003 USAMO |
Followed by 2005 USAMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.