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

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==Problem 10== | ==Problem 10== | ||

− | Given a point <math>P</math> in the coordinate plane, let <math>T_k(P) be the < | + | Given a point <math>P</math> in the coordinate plane, let <math>T_k(P)</math> be the <math>90\degree</math> rotation of <math>P</math> around the point <math>(2000-k,k)</math>. Let <math>P_0</math> be the point <math>(2007,0)</math> and <math>P_{n+1}=T_n(P_n)</math> for all integers <math>n\ge 0</math>. If <math>P_m</math> has a <math>y</math>-coordinate of <math>433</math>, what is <math>m</math>? |

[[Mock AIME 6 2006-2007 Problems/Problem 10|Solution]] | [[Mock AIME 6 2006-2007 Problems/Problem 10|Solution]] |

## Revision as of 14:28, 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 ?