Difference between revisions of "2005 USAMO Problems"
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= Day 2 = | = Day 2 = | ||
== Problem 4 == | == Problem 4 == | ||
− | + | Legs <math>L_1,L_2,L_3,L_4</math> of a square table each have length <math>n</math>, where <math>n</math> is a positive integer. For how many ordered 4-tuples <math>\left(k_1,k_2,k_3,k_4\right)</math> of nonnegative integers can we cut a piece of length <math>k_i</math> from the end of leg <math>L_i\ (i=1,2,3,4)</math> and still have a stable table? | |
+ | |||
+ | (The table is stable if it can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.) | ||
[[2005 USAMO Problems/Problem 4 | Solution]] | [[2005 USAMO Problems/Problem 4 | Solution]] | ||
== Problem 5 == | == Problem 5 == | ||
− | + | Let <math>n</math> be an integer greater than 1. Suppose <math>2n</math> points are given in the plane, no three of which are collinear. Suppose <math>n</math> of the given <math>2n</math> points are colored blue and the other <math>n</math> colored red. A line in the plane is called a ''balancing line'' if it passes through one blue and one red point and, for each side of the line, the number of blue points on that side is equal to the number of red points on the same side. | |
+ | |||
+ | Prove that there exist at least two balancing lines. | ||
[[2005 USAMO Problems/Problem 5 | Solution]] | [[2005 USAMO Problems/Problem 5 | Solution]] | ||
== Problem 6 == | == Problem 6 == | ||
− | {{ | + | For <math>m</math> a positive integer, let <math>s(m)</math> be the sum of the digits of <math>m</math>. For <math>n\ge 2</math>, let <math>f(n)</math> be the minimal <math>k</math> for which there exists a set <math>S</math> of <math>n</math> positive integers such that <math>s\left(\sum_{x\in X} x\right) = k</math> for any nonempty subset <math>X\subset S</math>. Prove that there are constants <math>0 < C_1 < C_2</math> with |
+ | <cmath> | ||
+ | C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n. | ||
+ | </cmath> | ||
[[2005 USAMO Problems/Problem 6 | Solution]] | [[2005 USAMO Problems/Problem 6 | Solution]] | ||
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= Resources = | = Resources = | ||
* [[USAMO Problems and Solutions]] | * [[USAMO Problems and Solutions]] | ||
− | * [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/ | + | * [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2005-ua/questions/Day1/05USAMOday1.pdf 2005 USAMO Day 1 Problems] |
− | * [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/ | + | * [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2005-ua/questions/Day2/05USAMOday2.pdf 2005 USAMO Day 2 Problems] |
+ | * [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2005-ua/Solutions/05SOL.pdf 2005 USAMO Solutions] | ||
* [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=2005 USAMO Problems on the Resources page] | * [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=2005 USAMO Problems on the Resources page] | ||
{{USAMO newbox|year=2005|before=[[2004 USAMO]]|after=2006 USAMO}} | {{USAMO newbox|year=2005|before=[[2004 USAMO]]|after=2006 USAMO}} |
Revision as of 13:40, 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
Legs of a square table each have length , where is a positive integer. For how many ordered 4-tuples of nonnegative integers can we cut a piece of length from the end of leg and still have a stable table?
(The table is stable if it can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)
Problem 5
Let be an integer greater than 1. Suppose points are given in the plane, no three of which are collinear. Suppose of the given points are colored blue and the other colored red. A line in the plane is called a balancing line if it passes through one blue and one red point and, for each side of the line, the number of blue points on that side is equal to the number of red points on the same side.
Prove that there exist at least two balancing lines.
Problem 6
For a positive integer, let be the sum of the digits of . For , let be the minimal for which there exists a set of positive integers such that for any nonempty subset . Prove that there are constants with
Resources
- USAMO Problems and Solutions
- 2005 USAMO Day 1 Problems
- 2005 USAMO Day 2 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 |