# Difference between revisions of "Mock AIME 6 2006-2007 Problems"

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==Problem 15== | ==Problem 15== | ||

− | For any finite sequence of positive integers <math>A=(a_1,a_2,\cdots,a_n)</math>, let <math>f(A)</math> be the sequence of the differences between consecutive terms of <math>A</math>. i.e. <math>f(A)=(a_2-a_1,a_3-a_2,\cdots,a_n-a_{n-1})</math>. Let <math>F^k(A)</math> denote <math>F</math> applied <math>k</math> times to <math>A</math>. If all of the sequences <math>A, | + | For any finite sequence of positive integers <math>A=(a_1,a_2,\cdots,a_n)</math>, let <math>f(A)</math> be the sequence of the differences between consecutive terms of <math>A</math>. i.e. <math>f(A)=(a_2-a_1,a_3-a_2,\cdots,a_n-a_{n-1})</math>. Let <math>F^k(A)</math> denote <math>F</math> applied <math>k</math> times to <math>A</math>. If all of the sequences <math>A, f(A), f^2(A),\cdots, f^{n-2}(A)</math> are strictly increasing and the only term of <math>f^{n01}(A)</math> is <math>1</math>, we call the sequence <math>A</math> <math>\textit{superpositive}</math>. How many sequences <math>A</math> with at least two terms and no terms exceeding <math>18</math> are <math>\textit{superpositive}</math>? |

[[Mock AIME 6 2006-2007 Problems/Problem 15|Solution]] | [[Mock AIME 6 2006-2007 Problems/Problem 15|Solution]] |

## Revision as of 14:43, 30 November 2014

## Contents

## Problem 1

Let be the sum of all positive integers of the form , where and are nonnegative integers that do not exceed . Find the remainder when is divided by .

## Problem 2

Draw in the diagonals of a regular octagon. What is the sum of all distinct angle measures, in degrees, formed by the intersections of the diagonals in the interior of the octagon?

## Problem 3

Alvin, Simon, and Theodore are running around a -meter circular track starting at different positions. Alvin is running in the opposite direction of Simon and Theodore. He is also the fastest, running twice as fast as Simon and three times as fast as Theodore. If Alvin meets Simon for the first time after running meters, and Simon meets Theodore for the first time after running meters, how far apart along the track (shorter distance) did Alvin and Theodore meet?

## Problem 4

Let be a set of points in the plane, no three of which lie on the same line. At most how many ordered triples of points in exist such that is obtuse?

## Problem 5

Let be the sum of the squares of the digits of . How many positive integers satisfy the inequality ?

## Problem 6

is a circle with radius and is a circle internally tangent to that passes through the center of . is a chord in of length tangent to at where . Given that where are positive integers and is not divisible by the square of any prime, what is ?

## Problem 7

Let and for all integers . How many more distinct complex roots does have than ?

## Problem 8

A sequence of positive reals defined by , , and for all integers . Given that and , find .

## Problem 9

is a triangle with integer side lengths. Extend beyond to point such that . Similarly, extend beyond to point such that and beyond to point such that . If triangles , , and all have the same area, what is the minimum possible area of triangle ?

## Problem 10

Given a point in the coordinate plane, let be the $90\degree$ (Error compiling LaTeX. ! Undefined control sequence.) rotation of around the point . Let be the point and for all integers . If has a -coordinate of , what is ?

## Problem 11

Each face of an octahedron is randomly colored blue or red. A caterpillar is on a vertex of the octahedron and wants to get to the opposite vertex by traversing the edges. The probability that it can do so without traveling along an edge that is shared by two faces of the same color is , where and are relatively prime positive integers. Find .

## Problem 12

Let be the largest positive rational solution to the equation for all integers . For each , let , where and are relatively prime positive integers. If what is the remainder when is divided by ?

## Problem 13

Consider two circles of different sizes that do not intersect. The smaller circle has center . Label the intersection of their common external tangents . A common internal tangent interesects the common external tangents at points and . Given that the radius of the larger circle is , , and , what is the square of the area of triangle ?

## Problem 14

A rational , where is a positive integer, is said to be if its base representation terminates. Let be the set of all rationals. The sum of all the elements in the union set $S_2\cupS_3\cup\cdots\cup S_{14}$ (Error compiling LaTeX. ! Undefined control sequence.) is , where and are relatively prime positive integers. Find .

## Problem 15

For any finite sequence of positive integers , let be the sequence of the differences between consecutive terms of . i.e. . Let denote applied times to . If all of the sequences are strictly increasing and the only term of is , we call the sequence . How many sequences with at least two terms and no terms exceeding are ?