Difference between revisions of "2006 USAMO Problems"

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== Day 1 ==
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= Day 1 =
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== Problem 1 ==
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Let <math>p</math> be a prime number and let <math>s</math> be an integer with <math>0<s<p</math>. Prove that there exist integers <math>m</math> and <math>n</math> with <math>0<m<n<p</math> and
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<cmath>\left\lbrace\frac{sm}{p}\right\rbrace<\left\lbrace\frac{sn}{p}\right\rbrace<\frac{s}{p}</cmath>
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if and only if <math>s</math> is not a divisor of <math>p-1</math>.
  
=== Problem 1 ===
+
Note: For <math>x</math> a real number, let <math>\lfloor x\rfloor</math> denote the greatest integer less than or equal to <math>x</math>, and let <math>\lbrace x\rbrace=x-\lfloor x\rfloor</math> denote the fractional part of <math>x</math>.
 
 
Let <math>\displaystyle p</math> be a prime number and let <math>\displaystyle s</math> be an integer with <math> \displaystyle 0 < s < p </math>. Prove that there exist integers <math>\displaystyle m</math> and <math>\displaystyle n</math> with <math>\displaystyle 0 < m < n < p</math> and
 
 
 
<center>
 
<math> \left\{ \frac{sm}{p} \right\} < \left\{ \frac{sn}{p} \right\} < \frac{s}{p} </math>
 
</center>
 
 
 
if and only if <math>\displaystyle s </math> is not a divisor of <math>\displaystyle p-1 </math>.
 
 
 
Note: For <math> \displaystyle x</math> a real number, let <math>\lfloor x \rfloor</math> denote the greatest integer less than or equal to <math>x</math>, and let <math>\{x\} = x - \lfloor x \rfloor</math> denote the fractional part of <math> \displaystyle x </math>.
 
  
 
[[2006 USAMO Problems/Problem 1 | Solution]]
 
[[2006 USAMO Problems/Problem 1 | Solution]]
  
=== Problem 2 ===
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== Problem 2 ==
 
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For a given positive integer <math>k</math> find, in terms of <math>k</math>, the minimum value of <math>N</math> for which there is a set of <math>2k+1</math> distinct positive integers that has sum greater than <math>N </math> but every subset of size <math>k</math> has sum at most <math>N/2</math>.
For a given positive integer <math> \displaystyle k </math> find, in terms of <math> \displaystyle k </math>, the minimum value of <math> \displaystyle N </math> for which there is a set of <math> \displaystyle 2k+1 </math> distinct positive integers that has sum greater than <math> \displaystyle N </math> but every subset of size <math> \displaystyle k </math> has sum at most <math>\displaystyle N/2 </math>.
 
  
 
[[2006 USAMO Problems/Problem 2 | Solution]]
 
[[2006 USAMO Problems/Problem 2 | Solution]]
  
=== Problem 3 ===
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== Problem 3 ==
 
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For integral <math>m</math>, let <math>p(m)</math> be the greatest prime divisor of <math>m</math>. By convention, we set <math> p(\pm1)=1</math> and <math>p(0)=\infty</math>. Find all polynomials <math>f</math> with integer coefficients such that the sequence <math>\lbrace p(f(n^2))-2n)\rbrace_{n\ge0}</math> is bounded above. (In particular, this requires <math>f(n^2)\neq0</math> for <math>n\ge0</math>.)
For integral <math>\displaystyle m </math>, let <math> \displaystyle p(m) </math> be the greatest prime divisor of <math> \displaystyle m </math>. By convention, we set <math> p(\pm 1)=1</math> and <math>p(0)=\infty</math>. Find all polynomials <math>\displaystyle f </math> with integer coefficients such that the sequence <math> \{ p(f(n^2))-2n) \} _{n\ge 0} </math> is bounded above. (In particular, this requires <math>f(n^2)\neq 0</math> for <math>n\ge 0</math>.)
 
  
 
[[2006 USAMO Problems/Problem 3 | Solution]]
 
[[2006 USAMO Problems/Problem 3 | Solution]]
  
== Day 2 ==
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= Day 2 =
 
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== Problem 4 ==
=== Problem 4 ===
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Find all positive integers <math>n</math> such that there are <math>k\ge 2</math> positive rational numbers <math>a_1,a_2,\ldots a_k</math> satisfying <math>a_1+a_2+\ldots+a_k=a_1\cdot a_2\cdots a_k=n</math>.
 
 
Find all positive integers <math> \displaystyle n</math> such that there are <math>k\ge 2</math> positive rational numbers <math>a_1,a_2,\ldots a_k</math> satisfying <math>a_1+a_2+...+a_k = a_1 \cdot a_2 \cdot \cdots a_k = n</math>.
 
  
 
[[2006 USAMO Problems/Problem 4 | Solution]]
 
[[2006 USAMO Problems/Problem 4 | Solution]]
  
=== Problem 5 ===
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== Problem 5 ==
 
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A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer <math>n </math>, then it can jump either to <math>n+1</math> or to <math>n+2^{m_n+1}</math> where <math>2^{m_n}</math> is the largest power of 2 that is a factor of <math>n</math>. Show that if <math>k\ge2</math> is a positive integer and <math>i</math> is a nonnegative integer, then the minimum number of jumps needed to reach <math>2^ik</math> is greater than the minimum number of jumps needed to reach <math>2^i</math>.
A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer <math> \displaystyle n </math>, then it can jump either to <math> \displaystyle n+1 </math> or to <math>n+2^{m_n+1}</math> where <math>2^{m_n}</math> is the largest power of 2 that is a factor of <math> \displaystyle n </math>. Show that if <math>k\ge 2</math> is a positive integer and <math> \displaystyle i </math> is a nonnegative integer, then the minimum number of jumps needed to reach <math> \displaystyle 2^i k </math> is greater than the minimum number of jumps needed to reach <math> \displaystyle 2^i </math>.
 
  
 
[[2006 USAMO Problems/Problem 5 | Solution]]
 
[[2006 USAMO Problems/Problem 5 | Solution]]
  
=== Problem 6 ===
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== Problem 6 ==
 
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Let <math>ABCD </math> be a quadrilateral, and let <math>E</math> and <math>F</math> be points on sides <math>AD</math> and <math>BC</math>, respectively, such that <math>AE/ED=BF/FC</math>.  Ray <math>FE</math> meets rays <math>BA</math> and <math>CD</math> at <math>S</math> and <math>T</math> respectively. Prove that the circumcircles of triangles <math>SAE</math>, <math>SBF</math>, <math>TCF</math>, and <math>TDE</math> pass through a common point.
Let <math> \displaystyle ABCD </math> be a quadrilateral, and let <math> \displaystyle  E </math> and <math> \displaystyle F </math> be points on sides <math> \displaystyle AD </math> and <math> \displaystyle BC </math>, respectively, such that <math> \displaystyle AE/ED = BF/FC </math>.  Ray <math> \displaystyle FE </math> meets rays <math> \displaystyle BA </math> and <math> \displaystyle CD </math> at <math> \displaystyle S </math> and <math> \displaystyle T </math> respectively. Prove that the circumcircles of triangles <math> \displaystyle SAE, SBF, TCF, </math> and <math> \displaystyle TDE </math> pass through a common point.
 
  
 
[[2006 USAMO Problems/Problem 6 | Solution]]
 
[[2006 USAMO Problems/Problem 6 | Solution]]
  
 
== Resources ==
 
== Resources ==
 +
*[[USAMO Problems and Solutions]]
 +
*[http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=2006 USAMO 2006 Problems on the Resources Page]
 +
*[http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2006-ua/2006usamoQ.pdf USAMO 2006 Questions Document]
 +
*[http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2006-ua/2006usamoS.pdf USAMO 2006 Solutions Document]
  
* [[2006 USAMO]]
+
{{USAMO newbox|year=2006|before=[[2005 USAMO]]|after=2007 USAMO}}
* [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=2006 USAMO 2006 Problems on the Resources Page]
 
* [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2006-ua/2006usamoQ.pdf USAMO 2006 Questions Document]
 
* [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2006-ua/2006usamoS.pdf USAMO 2006 Solutions Document]
 

Revision as of 01:51, 3 May 2008

Day 1

Problem 1

Let $p$ be a prime number and let $s$ be an integer with $0<s<p$. Prove that there exist integers $m$ and $n$ with $0<m<n<p$ and \[\left\lbrace\frac{sm}{p}\right\rbrace<\left\lbrace\frac{sn}{p}\right\rbrace<\frac{s}{p}\] if and only if $s$ is not a divisor of $p-1$.

Note: For $x$ a real number, let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$, and let $\lbrace x\rbrace=x-\lfloor x\rfloor$ denote the fractional part of $x$.

Solution

Problem 2

For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k+1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $N/2$.

Solution

Problem 3

For integral $m$, let $p(m)$ be the greatest prime divisor of $m$. By convention, we set $p(\pm1)=1$ and $p(0)=\infty$. Find all polynomials $f$ with integer coefficients such that the sequence $\lbrace p(f(n^2))-2n)\rbrace_{n\ge0}$ is bounded above. (In particular, this requires $f(n^2)\neq0$ for $n\ge0$.)

Solution

Day 2

Problem 4

Find all positive integers $n$ such that there are $k\ge 2$ positive rational numbers $a_1,a_2,\ldots a_k$ satisfying $a_1+a_2+\ldots+a_k=a_1\cdot a_2\cdots a_k=n$.

Solution

Problem 5

A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer $n$, then it can jump either to $n+1$ or to $n+2^{m_n+1}$ where $2^{m_n}$ is the largest power of 2 that is a factor of $n$. Show that if $k\ge2$ is a positive integer and $i$ is a nonnegative integer, then the minimum number of jumps needed to reach $2^ik$ is greater than the minimum number of jumps needed to reach $2^i$.

Solution

Problem 6

Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC$, respectively, such that $AE/ED=BF/FC$. Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$ respectively. Prove that the circumcircles of triangles $SAE$, $SBF$, $TCF$, and $TDE$ pass through a common point.

Solution

Resources

2006 USAMO (ProblemsResources)
Preceded by
2005 USAMO
Followed by
2007 USAMO
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All USAMO Problems and Solutions
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