Difference between revisions of "2011 AIME I Problems"
Line 39: | Line 39: | ||
== Problem 8 == | == Problem 8 == | ||
In triangle <math>ABC</math>, <math>BC = 23</math>, <math>CA = 27</math>, and <math>AB = 30</math>. Points <math>V</math> and <math>W</math> are on <math>\overline{AC}</math> with <math>V</math> on <math>\overline{AW}</math>, points <math>X</math> and <math>Y</math> are on <math>\overline{BC}</math> with <math>X</math> on <math>\overline{CY}</math>, and points <math>Z</math> and <math>U</math> are on <math>\overline{AB}</math> with <math>Z</math> on <math>\overline{BU}</math>. In addition, the points are positioned so that <math>\overline{UV} \parallel \overline{BC}</math>, <math>\overline{WX} \parallel \overline{AB}</math>, and <math>\overline{YZ} \parallel \overline{CA}</math>. Right angle folds are then made along <math>\overline{UV}</math>, <math>\overline{WX}</math>, and <math>\overline{YZ}</math>. The resulting figure is placed on a level floor to make a table with triangular legs. Let <math>h</math> be the maximum possible height of a table constructed from triangle <math>ABC</math> whose top is parallel to the floor. Then <math>h</math> can be written in the form <math>\frac{k \sqrt{m}}{n}</math>, where <math>k</math> and <math>n</math> are relatively prime positive integers and <math>m</math> is a positive integer that is not divisible by the square of any prime. Find <math>k + m + n</math>. | In triangle <math>ABC</math>, <math>BC = 23</math>, <math>CA = 27</math>, and <math>AB = 30</math>. Points <math>V</math> and <math>W</math> are on <math>\overline{AC}</math> with <math>V</math> on <math>\overline{AW}</math>, points <math>X</math> and <math>Y</math> are on <math>\overline{BC}</math> with <math>X</math> on <math>\overline{CY}</math>, and points <math>Z</math> and <math>U</math> are on <math>\overline{AB}</math> with <math>Z</math> on <math>\overline{BU}</math>. In addition, the points are positioned so that <math>\overline{UV} \parallel \overline{BC}</math>, <math>\overline{WX} \parallel \overline{AB}</math>, and <math>\overline{YZ} \parallel \overline{CA}</math>. Right angle folds are then made along <math>\overline{UV}</math>, <math>\overline{WX}</math>, and <math>\overline{YZ}</math>. The resulting figure is placed on a level floor to make a table with triangular legs. Let <math>h</math> be the maximum possible height of a table constructed from triangle <math>ABC</math> whose top is parallel to the floor. Then <math>h</math> can be written in the form <math>\frac{k \sqrt{m}}{n}</math>, where <math>k</math> and <math>n</math> are relatively prime positive integers and <math>m</math> is a positive integer that is not divisible by the square of any prime. Find <math>k + m + n</math>. | ||
+ | |||
+ | <center><asy> | ||
+ | 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); | ||
+ | </asy></center> | ||
[[2011 AIME I Problems/Problem 8|Solution]] | [[2011 AIME I Problems/Problem 8|Solution]] |
Revision as of 13:56, 18 March 2011
2011 AIME I (Answer Key) | AoPS Contest Collections | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Jar A contains four liters of a solution that is 45% acid. Jar B contains five liters of a solution that is 48% acid. Jar C contains one liter of a solution that is acid. From jar C, liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the end both jar A and jar B contain solutions that are 50% acid. Given that and are relatively prime positive integers, find .
Problem 2
In rectangle , and . Points and lie inside rectangle so that , , , , and line intersects segment . The length can be expressed in the form , where , , and are positive integers and is not divisible by the square of any prime. Find .
Problem 3
Let be the line with slope that contains the point , and let be the line perpendicular to line that contains the point . The original coordinate axes are erased, and line is made the -axis and line the -axis. In the new coordinate system, point is on the positive -axis, and point is on the positive -axis. The point with coordinates in the original system has coordinates in the new coordinate system. Find .
Problem 4
In triangle , , , and . The angle bisector of angle intersects at point , and the angle bisector of angle intersects at point . Let and be the feet of the perpendiculars from to and , respectively. Find .
Problem 5
The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits 1 through 9 in such a way that the sum of the numbers on every three consecutive vertices is a multiple of 3. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane. Find the number of distinguishable acceptable arrangements.
Problem 6
Suppose that a parabola has vertex and equation , where and is an integer. The minimum possible value of can be written in the form , where and are relatively prime positive integers. Find .
Problem 7
Find the number of positive integers for which there exist nonnegative integers , , \dots, 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 .
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 , 3, 5, 7, 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 .