# Difference between revisions of "2008 AIME II Problems"

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== Problem 11 == | == Problem 11 == | ||

− | {{ | + | In triangle <math>ABC</math>, <math>AB = AC = 100</math>, and <math>BC = 56</math>. Circle <math>P</math> has radius <math>16</math> and is tangent to <math>\bar{AC}</math> and <math>\bar{BC}</math>. Circle <math>Q</math> is externally tangent to circle <math>P</math> and is tangent to <math>\bar{AB}</math> and <math>\bar{BC}</math>. No point of circle <math>Q</math> lies outside of <math>\bigtriangleup\bar{ABC}</math>. The radius of circle <math>Q</math> can be expressed in the form <math>m - n\sqrt{k}</math>,where <math>m</math>, <math>n</math>, and <math>k</math> are positive integers and <math>k</math> is the product of distinct primes. Find <math>m +nk</math>. |

[[2008_AIME_II_Problems/Problem_11|Solution]] | [[2008_AIME_II_Problems/Problem_11|Solution]] |

## Revision as of 19:28, 3 April 2008

## Contents

## Problem 1

Let , where the additions and subtractions alternate in pairs. Find the remainder when is divided by .

## Problem 2

Rudolph bikes at a constant rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the -mile mark at exactly the same time. How many minutes has it taken them?

## Problem 3

A block of cheese in the shape of a rectangular solid measures cm by cm by cm. Ten slices are cut from the cheese. Each slice has a width of cm and is cut parallel to one face of the cheese. The individual slices are not necessarily parallel to each other. What is the maximum possible volume in cubic cm of the remaining block of cheese after ten slices have been cut off?

## Problem 4

There exist unique nonnegative integers and unique integers () with each either or such that Find .

## Problem 5

In trapezoid with , let and . Let , , and and be the midpoints of and , respectively. Find the length .

## Problem 6

The sequence is defined by The sequence is defined by Find .

## Problem 7

Let , , and be the three roots of the equation Find .

## Problem 8

Let . Find the smallest positive integer such that is an integer.

## Problem 9

A particle is located on the coordinate plane at . Define a *move* for the particle as a counterclockwise rotation of radians about the origin followed by a translation of units in the positive -direction. Given that the particle's position after moves is , find the greatest integer less than of equal to .

## Problem 10

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## Problem 11

In triangle , , and . Circle has radius and is tangent to and . Circle is externally tangent to circle and is tangent to and . No point of circle lies outside of . The radius of circle can be expressed in the form ,where , , and are positive integers and is the product of distinct primes. Find .

## Problem 12

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## Problem 13

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## Problem 14

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## Problem 15

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