Difference between revisions of "2011 AIME II Problems"
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== Problem 9 == | == Problem 9 == | ||
− | Let <math>x_1</math>, <math>x_2</math>, <math>\dots</math>, <math>x_6</math> be nonnegative real numbers such that <math>x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1</math>, and <math>x_1x_3x_5 + x_2x_4x_6 \ge \frac{1}{540}</math>. Let <math>p</math> and <math>q</math> be positive relatively prime integers such that <math>\frac{p}{q}</math> is the maximum possible value of <math>x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2</math>. Find <math>p + q</math>. | + | Let <math>x_1</math>, <math>x_2</math>, <math>\dots</math>, <math>x_6</math> be nonnegative real numbers such that <math>x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1</math>, and <math>x_1x_3x_5 + x_2x_4x_6 \ge {\scriptstyle\frac{1}{540}}</math>. Let <math>p</math> and <math>q</math> be positive relatively prime integers such that <math>\frac{p}{q}</math> is the maximum possible value of <math>x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2</math>. Find <math>p + q</math>. |
[[2011 AIME II Problems/Problem 9|Solution]] | [[2011 AIME II Problems/Problem 9|Solution]] |
Revision as of 16:46, 1 April 2011
2011 AIME II (Answer Key) | AoPS Contest Collections | ||
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Contents
Problem 1
Gary purchased a large bevarage, but only drank m/n of it, where m and n are relatively prime positive integers. If he had purchased half as much and drank twice as much, he would have wasted only 2/9 as much bevarage. Find m+n.
Problem 2
On square ABCD, point E lies on side AD and point F lies on side BC, so that BE=EF=FD=30. Find the area of the square ABCD.
Problem 3
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.
Problem 4
In triangle ABC, AB=(20/11)AC. The angle bisector of angle A intersects BC at point D, and point M is the midpoint of AD. Let P be the point of intersection of AC and the line BM. The ratio of CP to PA can be expresses in the form m/n, where m and n are relatively prime positive integers. Find m+n.
Problem 5
The sum of the first 2011 terms of a geometric sequence is 200. The sum of the first 4022 terms is 380. Find the sum of the first 6033 terms.
Problem 6
Define an ordered quadruple (a, b, c, d) as interesting if , and a+d>b+c. How many interesting ordered quadruples are there?
Problem 7
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 .
Problem 8
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 .
Problem 9
Let , , , be nonnegative real numbers such that , and . Let and be positive relatively prime integers such that is the maximum possible value of . Find .
Problem 10
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.
Problem 11
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 .
Problem 12
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 .
Problem 13
Point lies on the diagonal of square with $AP \gt CP$ (Error compiling LaTeX. ! Undefined control sequence.). Let and be the circumcenters of triangles and , respectively. Given that and $\angle O_1PO_2 = 120\textdegree$ (Error compiling LaTeX. ! Undefined control sequence.), then , where and are positive integers. Find .
Problem 14
There are permutations of such that for , divides for all integers with . Find the remainder when is divided by 1000.
Problem 15
Let . A real number is chosen at random from the interval . The probability that is equal to , where , , , , and are positive integers, and none of , , or is divisible by the square of a prime. Find .