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Difference between revisions of "2005 USAMO Problems"

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= Day 1 =
 
= Day 1 =
 
== Problem 1 ==
 
== Problem 1 ==
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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.
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(''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 ==
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Prove that the
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(''Răzvan Gelca'') Prove that the
 
system
 
system
 
<cmath>
 
<cmath>
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== Problem 3 ==
 
== Problem 3 ==
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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]]
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= Day 2 =
 
= Day 2 =
 
== Problem 4 ==
 
== Problem 4 ==
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{{solution}}
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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 ==
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{{solution}}
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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 ==
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{{solution}}
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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 =
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* [[USAMO Problems and Solutions]]
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* [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2005-ua/questions/Day1/05USAMOday1.pdf 2005 USAMO Day 1 Problems]
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* [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2006-ua/2006usamoQ.pdf 2005 USAMO Problems]
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* [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2005-ua/questions/Day2/05USAMOday2.pdf 2005 USAMO Day 2 Problems]
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* [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2006-ua/2006usamoS.pdf 2005 USAMO Solutions]
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* [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]
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{{USAMO newbox|year=2005|before=[[2004 USAMO]]|after=2006 USAMO}}
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{{USAMO newbox|year=2005|before=[[2004 USAMO]]|after=[[2006 USAMO]]}}
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{{MAA Notice}}

Latest revision as of 08:24, 14 May 2021

Day 1

Problem 1

(Zuming Feng) 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

(Răzvan Gelca) 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 $z$.

Solution

Problem 3

(Zuming Feng) 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.

Solution

Day 2

Problem 4

Legs $L_1,L_2,L_3,L_4$ of a square table each have length $n$, where $n$ is a positive integer. For how many ordered 4-tuples $\left(k_1,k_2,k_3,k_4\right)$ of nonnegative integers can we cut a piece of length $k_i$ from the end of leg $L_i\ (i=1,2,3,4)$ 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.)

Solution

Problem 5

Let $n$ be an integer greater than 1. Suppose $2n$ points are given in the plane, no three of which are collinear. Suppose $n$ of the given $2n$ points are colored blue and the other $n$ 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.

Solution

Problem 6

For $m$ a positive integer, let $s(m)$ be the sum of the digits of $m$. For $n\ge 2$, let $f(n)$ be the minimal $k$ for which there exists a set $S$ of $n$ positive integers such that $s\left(\sum_{x\in X} x\right) = k$ for any nonempty subset $X\subset S$. Prove that there are constants $0 < C_1 < C_2$ with \[C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.\]

Solution

Resources

2005 USAMO (Problems • Resources)
Preceded by
2004 USAMO 		Followed by
2006 USAMO
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

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