Difference between revisions of "2001 AIME II Problems"
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+ | {{AIME Problems|year=2001|n=II}} | ||
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== Problem 1 == | == Problem 1 == | ||
Let <math>N</math> be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of <math>N</math> forms a perfect square. What are the leftmost three digits of <math>N</math>? | Let <math>N</math> be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of <math>N</math> forms a perfect square. What are the leftmost three digits of <math>N</math>? | ||
+ | |||
[[2001 AIME II Problems/Problem 1|Solution]] | [[2001 AIME II Problems/Problem 1|Solution]] | ||
== Problem 2 == | == Problem 2 == | ||
− | Each of the 2001 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let <math>m</math> be the smallest number of students who could study both languages, and let <math>M</math> be the largest number of students who could study both languages. Find M-m. | + | Each of the 2001 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let <math>m</math> be the smallest number of students who could study both languages, and let <math>M</math> be the largest number of students who could study both languages. Find <math>M-m</math>. |
+ | |||
[[2001 AIME II Problems/Problem 2|Solution]] | [[2001 AIME II Problems/Problem 2|Solution]] | ||
== Problem 3 == | == Problem 3 == | ||
+ | Given that | ||
+ | <cmath> | ||
+ | \begin{align*}x_{1}&=211,\\ | ||
+ | x_{2}&=375,\\ | ||
+ | x_{3}&=420,\\ | ||
+ | x_{4}&=523,\ \text{and}\\ | ||
+ | x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4}\ \text{when}\ n\geq5, \end{align*} | ||
+ | </cmath> | ||
+ | find the value of <math>x_{531}+x_{753}+x_{975}</math>. | ||
[[2001 AIME II Problems/Problem 3|Solution]] | [[2001 AIME II Problems/Problem 3|Solution]] | ||
== Problem 4 == | == Problem 4 == | ||
+ | Let <math>R = (8,6)</math>. The lines whose equations are <math>8y = 15x</math> and <math>10y = 3x</math> contain points <math>P</math> and <math>Q</math>, respectively, such that <math>R</math> is the midpoint of <math>\overline{PQ}</math>. The length of <math>PQ</math> equals <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. | ||
[[2001 AIME II Problems/Problem 4|Solution]] | [[2001 AIME II Problems/Problem 4|Solution]] | ||
== Problem 5 == | == Problem 5 == | ||
+ | A set of positive numbers has the <math>triangle~property</math> if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets <math>\{4, 5, 6, \ldots, n\}</math> of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of <math>n</math>? | ||
[[2001 AIME II Problems/Problem 5|Solution]] | [[2001 AIME II Problems/Problem 5|Solution]] | ||
== Problem 6 == | == Problem 6 == | ||
+ | Square <math>ABCD</math> is inscribed in a circle. Square <math>EFGH</math> has vertices <math>E</math> and <math>F</math> on <math>\overline{CD}</math> and vertices <math>G</math> and <math>H</math> on the circle. The ratio of the area of square <math>EFGH</math> to the area of square <math>ABCD</math> can be expressed as <math>\frac {m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers and <math>m < n</math>. Find <math>10n + m</math>. | ||
[[2001 AIME II Problems/Problem 6|Solution]] | [[2001 AIME II Problems/Problem 6|Solution]] | ||
== Problem 7 == | == Problem 7 == | ||
+ | Let <math>\triangle{PQR}</math> be a right triangle with <math>PQ = 90</math>, <math>PR = 120</math>, and <math>QR = 150</math>. Let <math>C_{1}</math> be the inscribed circle. Construct <math>\overline{ST}</math> with <math>S</math> on <math>\overline{PR}</math> and <math>T</math> on <math>\overline{QR}</math>, such that <math>\overline{ST}</math> is perpendicular to <math>\overline{PR}</math> and tangent to <math>C_{1}</math>. Construct <math>\overline{UV}</math> with <math>U</math> on <math>\overline{PQ}</math> and <math>V</math> on <math>\overline{QR}</math> such that <math>\overline{UV}</math> is perpendicular to <math>\overline{PQ}</math> and tangent to <math>C_{1}</math>. Let <math>C_{2}</math> be the inscribed circle of <math>\triangle{RST}</math> and <math>C_{3}</math> the inscribed circle of <math>\triangle{QUV}</math>. The distance between the centers of <math>C_{2}</math> and <math>C_{3}</math> can be written as <math>\sqrt {10n}</math>. What is <math>n</math>? | ||
[[2001 AIME II Problems/Problem 7|Solution]] | [[2001 AIME II Problems/Problem 7|Solution]] | ||
== Problem 8 == | == Problem 8 == | ||
+ | A certain function <math>f</math> has the properties that <math>f(3x) = 3f(x)</math> for all positive real values of <math>x</math>, and that <math>f(x) = 1 - |x - 2|</math> for <math>1\leq x \leq 3</math>. Find the smallest <math>x</math> for which <math>f(x) = f(2001)</math>. | ||
[[2001 AIME II Problems/Problem 8|Solution]] | [[2001 AIME II Problems/Problem 8|Solution]] | ||
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== Problem 10 == | == Problem 10 == | ||
+ | How many positive integer multiples of 1001 can be expressed in the form <math>10^{j} - 10^{i}</math>, where <math>i</math> and <math>j</math> are integers and <math>0\leq i < j \leq 99</math>? | ||
[[2001 AIME II Problems/Problem 10|Solution]] | [[2001 AIME II Problems/Problem 10|Solution]] | ||
== Problem 11 == | == Problem 11 == | ||
+ | Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each <math>\frac {1}{3}</math>. The probability that Club Truncator will finish the season with more wins than losses is <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. | ||
[[2001 AIME II Problems/Problem 11|Solution]] | [[2001 AIME II Problems/Problem 11|Solution]] | ||
== Problem 12 == | == Problem 12 == | ||
+ | Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra <math>P_{i}</math> is defined recursively as follows: <math>P_{0}</math> is a regular tetrahedron whose volume is 1. To obtain <math>P_{i + 1}</math>, replace the midpoint triangle of every face of <math>P_{i}</math> by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of <math>P_{3}</math> is <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. | ||
[[2001 AIME II Problems/Problem 12|Solution]] | [[2001 AIME II Problems/Problem 12|Solution]] | ||
== Problem 13 == | == Problem 13 == | ||
+ | In quadrilateral <math>ABCD</math>, <math>\angle{BAD}\cong\angle{ADC}</math> and <math>\angle{ABD}\cong\angle{BCD}</math>, <math>AB = 8</math>, <math>BD = 10</math>, and <math>BC = 6</math>. The length <math>CD</math> may be written in the form <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. | ||
[[2001 AIME II Problems/Problem 13|Solution]] | [[2001 AIME II Problems/Problem 13|Solution]] | ||
== Problem 14 == | == Problem 14 == | ||
+ | There are <math>2n</math> complex numbers that satisfy both <math>z^{28} - z^{8} - 1 = 0</math> and <math>|z| = 1</math>. These numbers have the form <math>z_{m} = \cos\theta_{m} + i\sin\theta_{m}</math>, where <math>0\leq\theta_{1} < \theta_{2} < \ldots < \theta_{2n} < 360</math> and angles are measured in degrees. Find the value of <math>\theta_{2} + \theta_{4} + \ldots + \theta_{2n}</math>. | ||
[[2001 AIME II Problems/Problem 14|Solution]] | [[2001 AIME II Problems/Problem 14|Solution]] | ||
== Problem 15 == | == Problem 15 == | ||
+ | Let <math>EFGH</math>, <math>EFDC</math>, and <math>EHBC</math> be three adjacent square faces of a cube, for which <math>EC = 8</math>, and let <math>A</math> be the eighth vertex of the cube. Let <math>I</math>, <math>J</math>, and <math>K</math>, be the points on <math>\overline{EF}</math>, <math>\overline{EH}</math>, and <math>\overline{EC}</math>, respectively, so that <math>EI = EJ = EK = 2</math>. A solid <math>S</math> is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to <math>\overline{AE}</math>, and containing the edges, <math>\overline{IJ}</math>, <math>\overline{JK}</math>, and <math>\overline{KI}</math>. The surface area of <math>S</math>, including the walls of the tunnel, is <math>m + n\sqrt {p}</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>. | ||
[[2001 AIME II Problems/Problem 15|Solution]] | [[2001 AIME II Problems/Problem 15|Solution]] | ||
== See also == | == See also == | ||
+ | |||
+ | {{AIME box|year = 2001|n=II|before=[[2001 AIME I Problems]]|after=[[2002 AIME I Problems]]}} | ||
+ | |||
* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] | ||
* [[AIME Problems and Solutions]] | * [[AIME Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 21:12, 19 February 2019
2001 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Let be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of
forms a perfect square. What are the leftmost three digits of
?
Problem 2
Each of the 2001 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let be the smallest number of students who could study both languages, and let
be the largest number of students who could study both languages. Find
.
Problem 3
Given that
find the value of
.
Problem 4
Let . The lines whose equations are
and
contain points
and
, respectively, such that
is the midpoint of
. The length of
equals
, where
and
are relatively prime positive integers. Find
.
Problem 5
A set of positive numbers has the if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets
of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of
?
Problem 6
Square is inscribed in a circle. Square
has vertices
and
on
and vertices
and
on the circle. The ratio of the area of square
to the area of square
can be expressed as
where
and
are relatively prime positive integers and
. Find
.
Problem 7
Let be a right triangle with
,
, and
. Let
be the inscribed circle. Construct
with
on
and
on
, such that
is perpendicular to
and tangent to
. Construct
with
on
and
on
such that
is perpendicular to
and tangent to
. Let
be the inscribed circle of
and
the inscribed circle of
. The distance between the centers of
and
can be written as
. What is
?
Problem 8
A certain function has the properties that
for all positive real values of
, and that
for
. Find the smallest
for which
.
Problem 9
Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is , where
and
are relatively prime positive integers. Find
.
Problem 10
How many positive integer multiples of 1001 can be expressed in the form , where
and
are integers and
?
Problem 11
Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each . The probability that Club Truncator will finish the season with more wins than losses is
, where
and
are relatively prime positive integers. Find
.
Problem 12
Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra is defined recursively as follows:
is a regular tetrahedron whose volume is 1. To obtain
, replace the midpoint triangle of every face of
by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of
is
, where
and
are relatively prime positive integers. Find
.
Problem 13
In quadrilateral ,
and
,
,
, and
. The length
may be written in the form
, where
and
are relatively prime positive integers. Find
.
Problem 14
There are complex numbers that satisfy both
and
. These numbers have the form
, where
and angles are measured in degrees. Find the value of
.
Problem 15
Let ,
, and
be three adjacent square faces of a cube, for which
, and let
be the eighth vertex of the cube. Let
,
, and
, be the points on
,
, and
, respectively, so that
. A solid
is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to
, and containing the edges,
,
, and
. The surface area of
, including the walls of the tunnel, is
, where
,
, and
are positive integers and
is not divisible by the square of any prime. Find
.
See also
2001 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2001 AIME I Problems |
Followed by 2002 AIME I Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- 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.