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 18:56, 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
![$\frac{sm}{p}$](http://latex.artofproblemsolving.com/3/e/d/3ed232b94ed9ff8da7fb5d9f8b5593d9c1e6d6f1.png)
![$\frac{sn}{p}$](http://latex.artofproblemsolving.com/9/6/8/968de9fced2c629642619fd43e91865a80b56c6f.png)
![${\frac{s}{p}}$](http://latex.artofproblemsolving.com/8/b/0/8b019b237f57073f7b250eaede1699dfcfe93eff.png)
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.