2011 AIME II真题
Gary purchased a large beverage, but only drank of it, where and are relatively prime positive integers. If he had purchased half as much and drunk twice as much, he would have wasted only as much beverage. Find .
On square , point lies on side and point lies on side , so that . Find the area of the square .
The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.
In triangle , . The angle bisector of angle intersects at point , and point is the midpoint of . Let be the point of intersection of and the line . The ratio of to can be expressed in the form , where and are relatively prime positive integers. Find .
The sum of the first terms of a geometric sequence is . The sum of the first terms is . Find the sum of the first terms.
Define an ordered quadruple of integers as interesting if , and . How many interesting ordered quadruples are there?
Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves is equal to the number of marbles whose right hand neighbor is the other color. An example of such an arrangement is GGRRRGGRG. Let be the maximum number of red marbles for which such an arrangement is possible, and let be the number of ways he can arrange the marbles to satisfy the requirement. Find the remainder when is divided by .
Let , , , , be the 12 zeroes of the polynomial . For each , let be one of or . Then the maximum possible value of the real part of can be written as where and are positive integers. Find .
Let , , , be nonnegative real numbers such that , and. Let and be positive relatively prime integers such that is the maximum possible value of . Find .
A circle with center has radius 25. Chord of length 30 and chord of length 14 intersect at point . The distance between the midpoints of the two chords is 12. The quantity can be represented as , where and are relatively prime positive integers. Find the remainder when is divided by 1000.
Let be the matrix with entries as follows: for , ; for , ; all other entries in are zero. Let be the determinant of matrix . Then can be represented as , where and are relatively prime positive integers. Find . Note: The determinant of the matrix is , and the determinant of the matrix ; for , the determinant of an matrix with first row or first column is equal to , where is the determinant of the matrix formed by eliminating the row and column containing .
Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be , where and are relatively prime positive integers. Find.
Point lies on the diagonal of square with . Let and be the circumcenters of triangles and , respectively. Given that and , then , where and are positive integers. Find .
There are permutations of such that for , divides for all integers with . Find the remainder when is divided by 1000.
Let . A real number is chosen at random from the interval . The probability that is equal to , where , , , , and are positive integers. Find .