Difference between revisions of "2011 AIME II Problems"
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Note: All questions with a star after the problem number are not yet the correct problem, as I copy/pasted the format from the 2011 AIME I page. | Note: All questions with a star after the problem number are not yet the correct problem, as I copy/pasted the format from the 2011 AIME I page. | ||
− | == Problem 1 | + | == 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''. | |
[[2011 AIME II Problems/Problem 1|Solution]] | [[2011 AIME II Problems/Problem 1|Solution]] | ||
− | == Problem 2 | + | == 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. | |
[[2011 AIME II Problems/Problem 2|Solution]] | [[2011 AIME II Problems/Problem 2|Solution]] | ||
− | == Problem 3 | + | == 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. | |
[[2011 AIME II Problems/Problem 3|Solution]] | [[2011 AIME II Problems/Problem 3|Solution]] | ||
− | == Problem 4 | + | == Problem 4 == |
− | In triangle | + | 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. |
[[2011 AIME II Problems/Problem 4|Solution]] | [[2011 AIME II Problems/Problem 4|Solution]] | ||
− | == Problem 5 | + | == Problem 5 == |
− | The | + | 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. |
[[2011 AIME II Problems/Problem 5|Solution]] | [[2011 AIME II Problems/Problem 5|Solution]] | ||
− | == Problem 6 | + | == Problem 6 == |
− | + | Define an ordered quadruple (a, b, c, d) as interesting if <math>1 \le a<b<c<d \le 10</math>, and a+d>b+c. How many ordered quadruples are there? | |
[[2011 AIME II Problems/Problem 6|Solution]] | [[2011 AIME II Problems/Problem 6|Solution]] |
Revision as of 09:02, 31 March 2011
2011 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Note: All questions with a star after the problem number are not yet the correct problem, as I copy/pasted the format from the 2011 AIME I page.
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 ordered quadruples are there?
Problem 7*
Find the number of positive integers for which there exist nonnegative integers , , , such that
Problem 8*
In triangle , , , and . Points and are on with on , points and are on with on , and points and are on with on . In addition, the points are positioned so that , , and . Right angle folds are then made along , , and . The resulting figure is placed on a level floor to make a table with triangular legs. Let be the maximum possible height of a table constructed from triangle whose top is parallel to the floor. Then can be written in the form , where and are relatively prime positive integers and is a positive integer that is not divisible by the square of any prime. Find .
unitsize(1 cm); pair translate; pair[] A, B, C, U, V, W, X, Y, Z; A[0] = (1.5,2.8); B[0] = (3.2,0); C[0] = (0,0); U[0] = (0.69*A[0] + 0.31*B[0]); V[0] = (0.69*A[0] + 0.31*C[0]); W[0] = (0.69*C[0] + 0.31*A[0]); X[0] = (0.69*C[0] + 0.31*B[0]); Y[0] = (0.69*B[0] + 0.31*C[0]); Z[0] = (0.69*B[0] + 0.31*A[0]); translate = (7,0); A[1] = (1.3,1.1) + translate; B[1] = (2.4,-0.7) + translate; C[1] = (0.6,-0.7) + translate; U[1] = U[0] + translate; V[1] = V[0] + translate; W[1] = W[0] + translate; X[1] = X[0] + translate; Y[1] = Y[0] + translate; Z[1] = Z[0] + translate; draw (A[0]--B[0]--C[0]--cycle); draw (U[0]--V[0],dashed); draw (W[0]--X[0],dashed); draw (Y[0]--Z[0],dashed); draw (U[1]--V[1]--W[1]--X[1]--Y[1]--Z[1]--cycle); draw (U[1]--A[1]--V[1],dashed); draw (W[1]--C[1]--X[1]); draw (Y[1]--B[1]--Z[1]); dot("$A$",A[0],N); dot("$B$",B[0],SE); dot("$C$",C[0],SW); dot("$U$",U[0],NE); dot("$V$",V[0],NW); dot("$W$",W[0],NW); dot("$X$",X[0],S); dot("$Y$",Y[0],S); dot("$Z$",Z[0],NE); dot(A[1]); dot(B[1]); dot(C[1]); dot("$U$",U[1],NE); dot("$V$",V[1],NW); dot("$W$",W[1],NW); dot("$X$",X[1],dir(-70)); dot("$Y$",Y[1],dir(250)); dot("$Z$",Z[1],NE); (Error making remote request. Unexpected URL sent back)
Problem 9*
Suppose is in the interval and . Find .
Problem 10*
The probability that a set of three distinct vertices chosen at random from among the vertices of a regular -gon determine an obtuse triangle is . Find the sum of all possible values of .
Problem 11*
Let be the set of all possible remainders when a number of the form , a nonnegative integer, is divided by 1000. Let be the sum of the elements in . Find the remainder when is divided by 1000.
Problem 12*
Six men and some number of women stand in a line in random order. Let be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that does not exceed 1 percent.
Problem 13*
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled . The three vertices adjacent to vertex are at heights 10, 11, and 12 above the plane. The distance from vertex to the plane can be expressed as , where , , and are positive integers. Find .
Problem 14*
Let be a regular octagon. Let , , , and be the midpoints of sides , , , and , respectively. For , ray is constructed from towards the interior of the octagon such that , , , and . Pairs of rays and , and , and , and and meet at , , , respectively. If , then can be written in the form , where and are positive integers. Find .
Problem 15*
For some integer , the polynomial has the three integer roots , , and . Find .