# Difference between revisions of "2010 USAJMO Problems"

m (→Problem 2) |
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three properties: | three properties: | ||

<ol style="list-style-type:lower-alpha"> | <ol style="list-style-type:lower-alpha"> | ||

− | <li> | + | <li> <math>x_1 < x_2 < \cdots <x_{n-1}</math>; |

− | <li> | + | <li> <math>x_i +x_{n-i} = 2n</math> for all <math>i=1,2,\ldots,n-1</math>; |

− | <li> | + | <li> given any two indices <math>i</math> and <math>j</math> (not necessarily distinct) |

for which <math>x_i + x_j < 2n</math>, there is an index <math>k</math> such | for which <math>x_i + x_j < 2n</math>, there is an index <math>k</math> such | ||

that <math>x_i+x_j = x_k</math>. | that <math>x_i+x_j = x_k</math>. |

## Revision as of 08:18, 12 October 2012

## Contents

## Day 1

### Problem 1

A permutation of the set of positive integers is a sequence such that each element of appears precisely one time as a term of the sequence. For example, is a permutation of . Let be the number of permutations of for which is a perfect square for all . Find with proof the smallest such that is a multiple of .

### Problem 2

Let be an integer. Find, with proof, all sequences of positive integers with the following three properties:

- ;
- for all ;
- given any two indices and (not necessarily distinct) for which , there is an index such that .

### Problem 3

Let be a convex pentagon inscribed in a semicircle of diameter . Denote by the feet of the perpendiculars from onto lines , respectively. Prove that the acute angle formed by lines and is half the size of , where is the midpoint of segment .

## Day 2

### Problem 4

A triangle is called a parabolic triangle if its vertices lie on a parabola . Prove that for every nonnegative integer , there is an odd number and a parabolic triangle with vertices at three distinct points with integer coordinates with area .

### Problem 5

Two permutations and of the numbers are said to intersect if for some value of in the range . Show that there exist permutations of the numbers such that any other such permutation is guaranteed to intersect at least one of these permutations.

### Problem 6

Let be a triangle with . Points and lie on sides and , respectively, such that and . Segments and meet at . Determine whether or not it is possible for segments to all have integer lengths.