# Difference between revisions of "2006 USAMO Problems"

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== Day 1 == | == Day 1 == | ||

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=== Problem 1 === | === Problem 1 === | ||

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− | + | 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]] | |

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=== Problem 2 === | === Problem 2 === | ||

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− | [[2006 USAMO Problems/Problem 2|Solution]] | + | 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>. |

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+ | [[2006 USAMO Problems/Problem 2 | Solution]] | ||

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=== Problem 3 === | === Problem 3 === | ||

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− | <math>(p(f(n^2))-2n)_{n\ge 0}</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>.) |

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+ | [[2006 USAMO Problems/Problem 3 | Solution]] | ||

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+ | == Day 2 == | ||

+ | |||

+ | === Problem 4 === | ||

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+ | 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]] | ||

+ | |||

+ | === Problem 5 === | ||

− | is | + | 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 | + | [[2006 USAMO Problems/Problem 5 | Solution]] |

− | + | === Problem 6 === | |

− | === Problem | ||

− | |||

− | + | 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. | |

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− | [[2006 USAMO Problems/Problem | + | [[2006 USAMO Problems/Problem 6 | Solution]] |

− | === | + | == Resources == |

− | |||

− | [[2006 USAMO Problems/ | + | * [[2006 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 20:05, 1 September 2006

## Contents

## Day 1

### Problem 1

Let be a prime number and let be an integer with . Prove that there exist integers and with and

if and only if is not a divisor of .

Note: For a real number, let denote the greatest integer less than or equal to , and let denote the fractional part of .

### Problem 2

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

### Problem 3

For integral , let be the greatest prime divisor of . By convention, we set and . Find all polynomials with integer coefficients such that the sequence is bounded above. (In particular, this requires for .)

## Day 2

### Problem 4

Find all positive integers such that there are positive rational numbers satisfying .

### 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 , then it can jump either to or to where is the largest power of 2 that is a factor of . Show that if is a positive integer and is a nonnegative integer, then the minimum number of jumps needed to reach is greater than the minimum number of jumps needed to reach .

### Problem 6

Let be a quadrilateral, and let and be points on sides and , respectively, such that . Ray meets rays and at and respectively. Prove that the circumcircles of triangles and pass through a common point.