2009 AIME II真题
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 stripe. 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.
Suppose that , , and are positive real numbers such that , , and . Find
In rectangle , . Let be the midpoint of . Given that line and line are perpendicular, find the greatest integer less than .
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 .
Equilateral triangle is inscribed in circle , which has radius . Circle with radius is internally tangent to circle at one vertex of . Circles and , both with radius , are internally tangent to circle at the other two vertices of . Circles , , and are all externally tangent to circle , which has radius , where and are relatively prime positive integers. Find .
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 .
Define to be for odd and for even. When is expressed as a fraction in lowest terms, its denominator is with odd. Find .
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 .
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 .
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 .
For certain pairs of positive integers with there are exactly distinct positive integers such that . Find the sum of all possible values of the product .
From the set of integers , choose pairs with so that no two pairs have a common element. Suppose that all the sums are distinct and less than or equal to . Find the maximum possible value of .
Let and be the endpoints of a semicircular arc of radius . The arc is divided into seven congruent arcs by six equally spaced points . All chords of the form or are drawn. Let be the product of the lengths of these twelve chords. Find the remainder when is divided by .
The sequence satisfies and for . Find the greatest integer less than or equal to .
Let be a diameter of a circle with diameter . Let and be points on one of the semicircular arcs determined by such that is the midpoint of the semicircle and . Point lies on the other semicircular arc. Let be the length of the line segment whose endpoints are the intersections of diameter with the chords and . The largest possible value of can be written in the form , where , , and are positive integers and is not divisible by the square of any prime. Find .
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