# Difference between revisions of "2006 USAMO Problems"

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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>\{x\} = x - \lfloor x \rfloor</math> denote the fractional part of x. | 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>\{x\} = x - \lfloor x \rfloor</math> denote the fractional part of x. | ||

− | [[2006 USAMO Problems | + | [[2006 USAMO Problems/Problem 1|Solution]] |

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

For a given positive integer '''k''' find, in terms of '''k''', 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 '''k''' has sum at most <math>\frac{N}{2}</math>. | For a given positive integer '''k''' find, in terms of '''k''', 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 '''k''' has sum at most <math>\frac{N}{2}</math>. | ||

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

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

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(\pm 1)=1</math> and <math>p(0)=\infty</math>. Find all polynomial <math>f</math> with integer coefficients such that the sequence | 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(\pm 1)=1</math> and <math>p(0)=\infty</math>. Find all polynomial <math>f</math> with integer coefficients such that the sequence | ||

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is bounded above. (In particular, this requires <math>f(n^2)\neq 0</math> for <math>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 | + | [[2006 USAMO Problems/Problem 3|Solution]] |

== Day 2 == | == Day 2 == | ||

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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+...+a_k=a_1\cdot a_2\cdots a_k=n.</math> | 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+...+a_k=a_1\cdot a_2\cdots a_k=n.</math> | ||

− | [[2006 USAMO Problems | + | [[2006 USAMO Problems/Problem 4|Solution]] |

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

A mathematical frog jumps along the number line. The frog starts at <math>1</math>, 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 <math>2</math> that is a factor of <math>n</math>. Show that if <math>k/ge 2</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 <math>1</math>, 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 <math>2</math> that is a factor of <math>n</math>. Show that if <math>k/ge 2</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> | ||

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

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

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>\frac{AE}{ED}=\frac{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, SBF, TCF,</math> and <math>TDE</math> pass through a common point. | 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>\frac{AE}{ED}=\frac{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, SBF, TCF,</math> and <math>TDE</math> pass through a common point. | ||

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

## Revision as of 17:56, 9 July 2006

## Contents

## Day 1

### Problem 1

Let be a prime number and let be an integer with **.** Prove that there exists 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 x.

### Problem 2

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

### Problem 3

For integral , let be the greatest prime divisor of . By convention, we set and . Find all polynomial with integer coefficients such that the sequence

is bounded above. (In particular, this requires for

## Day 2

### Problem 1

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

### Problem 2

A mathematical frog jumps along the number line. The frog starts at , 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 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 3

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.