Difference between revisions of "2005 USAMO Problems"
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= Day 1 = | = Day 1 = | ||
== Problem 1 == | == Problem 1 == | ||
− | 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. | + | (''Zuming Feng'') 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]] | ||
== Problem 2 == | == Problem 2 == | ||
− | Prove that the | + | (''Răzvan Gelca'') Prove that the |
system | system | ||
<cmath> | <cmath> | ||
Line 19: | Line 19: | ||
== Problem 3 == | == Problem 3 == | ||
− | 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. | + | (''Zuming Feng'') 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]] | [[2005 USAMO Problems/Problem 3 | Solution]] | ||
Line 25: | Line 25: | ||
= 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]] | ||
= Resources = | = Resources = | ||
− | * [ | + | * [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/ | + | * [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= | + | {{USAMO newbox|year=2005|before=[[2004 USAMO]]|after=[[2006 USAMO]]}} |
+ | {{MAA Notice}} |
Latest revision as of 09:24, 14 May 2021
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
- 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.