Difference between revisions of "2011 AIME I Problems"
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== Problem 11 == | == Problem 11 == | ||
Let <math>R</math> be the set of all possible remainders when a number of the form <math>2^n</math>, <math>n</math> a nonnegative integer, is divided by 1000. Let <math>S</math> be the sum of the elements in <math>R</math>. Find the remainder when <math>S</math> is divided by 1000. | Let <math>R</math> be the set of all possible remainders when a number of the form <math>2^n</math>, <math>n</math> a nonnegative integer, is divided by 1000. Let <math>S</math> be the sum of the elements in <math>R</math>. Find the remainder when <math>S</math> is divided by 1000. | ||
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[[2011 AIME I Problems/Problem 11|Solution]] | [[2011 AIME I Problems/Problem 11|Solution]] | ||
Revision as of 17:18, 8 May 2020
2011 AIME I (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 |
Contents
Problem 1
Jar A contains four liters of a solution that is acid. Jar B contains five liters of a solution that is
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
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
,
,
,
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
.
See also
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.