2009 AIME II Problems
2009 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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Contents
Problem 1
Before starting to paint, Bill had ounces of blue paint, ounces of red paint, and ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stirpe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left.
Problem 2
Suppose that , , and are positive real numbers such that , , and . Find
Problem 3
In rectangle , . Let be the midpoint of . Given that line and line are perpendicular, find the greatest integer less than .
Problem 4
A group of children held a grape-eating contest. When the contest was over, the winner had eaten grapes, and the child in -th place had eaten grapes. The total number of grapes eaten in the contest was . Find the smallest possible value of .
Problem 5
Problem 6
Let be the number of five-element subsets that can be chosen from the set of the first natural numbers so that at least two of the five numbers are consecutive. Find the remainder when is divided by .
Problem 7
Define to be for odd and for even. When is expressed as a fraction in lowest terms, its denominator is with odd. Find .
Problem 8
Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let and be relatively prime positive integers such that is the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. Find .
Problem 9
Let be the number of solutions in positive integers to the equation , and let be the number of solutions in positive integers to the equation . Find the remainder when is divided by .
Problem 10
Four lighthouses are located at points , , , and . The lighthouse at is kilometers from the lighthouse at , the lighthouse at is kilometers from the lighthouse at , and the lighthouse at is kilometers from the lighthouse at . To an observer at , the angle determined by the lights at and and the angle determined by the lights at and are equal. To an observer at , the angle determined by the lights at and and the angle determined by the lights at and are equal. The number of kilometers from to is given by , where , , and are relatively prime positive integers, and is not divisible by the square of any prime. Find .