Difference between revisions of "2002 AIME I Problems"
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+ | {{AIME Problems|year=2002|n=I}} | ||
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== Problem 1 == | == Problem 1 == | ||
Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-to-left) is <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-to-left) is <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
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The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as <math>\frac{1}{2}\left(\sqrt{p}-q\right)</math>, where <math>p</math> and <math>q</math> are positive integers. Find <math>p+q</math>. | The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as <math>\frac{1}{2}\left(\sqrt{p}-q\right)</math>, where <math>p</math> and <math>q</math> are positive integers. Find <math>p+q</math>. | ||
− | < | + | <asy> |
+ | size(250);real x=sqrt(3); | ||
+ | int i; | ||
+ | draw(origin--(14,0)--(14,2+2x)--(0,2+2x)--cycle); | ||
+ | for(i=0; i<7; i=i+1) { | ||
+ | draw(Circle((2*i+1,1), 1)^^Circle((2*i+1,1+2x), 1)); | ||
+ | } | ||
+ | for(i=0; i<6; i=i+1) { | ||
+ | draw(Circle((2*i+2,1+x), 1)); | ||
+ | } | ||
+ | </asy> | ||
[[2002 AIME I Problems/Problem 2|Solution]] | [[2002 AIME I Problems/Problem 2|Solution]] | ||
== Problem 3 == | == Problem 3 == | ||
− | Jane is 25 years old. Dick is older than Jane. In <math>n</math> years, where n is a positive integer, Dick's age and Jane's age will both be two-digit | + | Jane is 25 years old. Dick is older than Jane. In <math>n</math> years, where <math>n</math> is a positive integer, Dick's age and Jane's age will both be two-digit numbers and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let <math>d</math> be Dick's present age. How many ordered pairs of positive integers <math>(d,n)</math> are possible? |
[[2002 AIME I Problems/Problem 3|Solution]] | [[2002 AIME I Problems/Problem 3|Solution]] | ||
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The solutions to the system of equations | The solutions to the system of equations | ||
− | < | + | <cmath> |
− | + | \begin{align*} | |
+ | \log_{225}{x}+\log_{64}{y} = 4\\ | ||
+ | \log_{x}{225}- \log_{y}{64} = 1 | ||
+ | \end{align*} | ||
+ | </cmath> | ||
are <math>(x_1,y_1)</math> and <math>(x_2, y_2)</math>. Find <math>\log_{30}{(x_1y_1x_2y_2)}</math>. | are <math>(x_1,y_1)</math> and <math>(x_2, y_2)</math>. Find <math>\log_{30}{(x_1y_1x_2y_2)}</math>. | ||
Line 36: | Line 52: | ||
== Problem 7 == | == Problem 7 == | ||
− | The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers <math>x</math>, <math>y</math>, and <math>r</math> with <math>|x|>|y|</math>, <center><math>(x+y)^r=x^r+rx^{r-1}+\frac{r(r-1)}2x^{r-2}y^2+\frac{r(r-1)(r-2)}{3!}x^{r-3}y^3+\cdots</math></center> | + | The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers <math>x</math>, <math>y</math>, and <math>r</math> with <math>|x|>|y|</math>, <center><math>(x+y)^r=x^r+rx^{r-1}y^1+\frac{r(r-1)}2x^{r-2}y^2+\frac{r(r-1)(r-2)}{3!}x^{r-3}y^3+\cdots</math></center> |
What are the first three digits to the right of the decimal point in the decimal representation of <math>\left(10^{2002}+1\right)^{10/7}</math>? | What are the first three digits to the right of the decimal point in the decimal representation of <math>\left(10^{2002}+1\right)^{10/7}</math>? | ||
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== Problem 8 == | == Problem 8 == | ||
− | Find the smallest integer k for which the conditions | + | Find the smallest integer <math>k</math> for which the conditions |
(1) <math>a_1, a_2, a_3, \ldots</math> is a nondecreasing sequence of positive integers | (1) <math>a_1, a_2, a_3, \ldots</math> is a nondecreasing sequence of positive integers | ||
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In the diagram below, angle <math>ABC</math> is a right angle. Point <math>D</math> is on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects angle <math>CAB</math>. Points <math>E</math> and <math>F</math> are on <math>\overline{AB}</math> and <math>\overline{AC}</math>, respectively, so that <math>AE=3</math> and <math>AF=10</math>. Given that <math>EB=9</math> and <math>FC=27</math>, find the integer closest to the area of quadrilateral <math>DCFG</math>. | In the diagram below, angle <math>ABC</math> is a right angle. Point <math>D</math> is on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects angle <math>CAB</math>. Points <math>E</math> and <math>F</math> are on <math>\overline{AB}</math> and <math>\overline{AC}</math>, respectively, so that <math>AE=3</math> and <math>AF=10</math>. Given that <math>EB=9</math> and <math>FC=27</math>, find the integer closest to the area of quadrilateral <math>DCFG</math>. | ||
− | < | + | <asy> |
+ | size(250); | ||
+ | pair A=(0,12), E=(0,8), B=origin, C=(24*sqrt(2),0), D=(6*sqrt(2),0), F=A+10*dir(A--C), G=intersectionpoint(E--F, A--D); | ||
+ | draw(A--B--C--A--D^^E--F); | ||
+ | pair point=G+1*dir(250); | ||
+ | label("$A$", A, dir(point--A)); | ||
+ | label("$B$", B, dir(point--B)); | ||
+ | label("$C$", C, dir(point--C)); | ||
+ | label("$D$", D, dir(point--D)); | ||
+ | label("$E$", E, dir(point--E)); | ||
+ | label("$F$", F, dir(point--F)); | ||
+ | label("$G$", G, dir(point--G)); | ||
+ | markscalefactor=0.1; | ||
+ | draw(rightanglemark(A,B,C)); | ||
+ | label("10", A--F, dir(90)*dir(A--F)); | ||
+ | label("27", F--C, dir(90)*dir(F--C)); | ||
+ | label("3", (0,10), W); | ||
+ | label("9", (0,4), W); | ||
+ | </asy> | ||
[[2002 AIME I Problems/Problem 10|Solution]] | [[2002 AIME I Problems/Problem 10|Solution]] | ||
Line 78: | Line 112: | ||
== Problem 13 == | == Problem 13 == | ||
− | In triangle <math>ABC</math> the medians <math>\overline{AD}</math> and <math>\overline{CE}</math> have lengths 18 and 27, respectively, and <math>AB = 24</math>. Extend <math>\overline{CE}</math> to intersect the circumcircle of <math>ABC</math> at <math>F</math>. The area of | + | In triangle <math>ABC</math> the medians <math>\overline{AD}</math> and <math>\overline{CE}</math> have lengths 18 and 27, respectively, and <math>AB = 24</math>. Extend <math>\overline{CE}</math> to intersect the circumcircle of <math>ABC</math> at <math>F</math>. The area of triangle <math>AFB</math> is <math>m\sqrt {n}</math>, where <math>m</math> and <math>n</math> are positive integers and <math>n</math> is not divisible by the square of any prime. Find <math>m + n</math>. |
[[2002 AIME I Problems/Problem 13|Solution]] | [[2002 AIME I Problems/Problem 13|Solution]] | ||
== Problem 14 == | == Problem 14 == | ||
− | A set <math>\mathcal{S}</math> of distinct positive integers has the following property: for every integer <math>x</math> in <math>\mathcal{S},</math> the arithmetic mean of the set of values obtained by deleting <math>x</math> from <math>\mathcal{S}</math> is an integer. Given that 1 belongs to <math>\mathcal{S}</math> and that 2002 is the largest element of <math>\mathcal{S},</math> what is the | + | A set <math>\mathcal{S}</math> of distinct positive integers has the following property: for every integer <math>x</math> in <math>\mathcal{S},</math> the arithmetic mean of the set of values obtained by deleting <math>x</math> from <math>\mathcal{S}</math> is an integer. Given that 1 belongs to <math>\mathcal{S}</math> and that 2002 is the largest element of <math>\mathcal{S},</math> what is the greatest number of elements that <math>\mathcal{S}</math> can have? |
[[2002 AIME I Problems/Problem 14|Solution]] | [[2002 AIME I Problems/Problem 14|Solution]] | ||
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== See also == | == See also == | ||
+ | |||
+ | {{AIME box|year = 2002|n=I|before=[[2001 AIME II Problems]]|after=[[2002 AIME II 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 21:09, 27 July 2023
2002 AIME I (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
Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right-to-left) is , where
and
are relatively prime positive integers. Find
.
Problem 2
The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as , where
and
are positive integers. Find
.
Problem 3
Jane is 25 years old. Dick is older than Jane. In years, where
is a positive integer, Dick's age and Jane's age will both be two-digit numbers and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let
be Dick's present age. How many ordered pairs of positive integers
are possible?
Problem 4
Consider the sequence defined by for
. Given that
, for positive integers
and
with
, find
.
Problem 5
Let be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set
?
Problem 6
The solutions to the system of equations
are
and
. Find
.
Problem 7
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers ,
, and
with
,
![$(x+y)^r=x^r+rx^{r-1}y^1+\frac{r(r-1)}2x^{r-2}y^2+\frac{r(r-1)(r-2)}{3!}x^{r-3}y^3+\cdots$](http://latex.artofproblemsolving.com/6/7/1/671e1dc221b4e56242537edaaef7ca5d1665b814.png)
What are the first three digits to the right of the decimal point in the decimal representation of ?
Problem 8
Find the smallest integer for which the conditions
(1) is a nondecreasing sequence of positive integers
(2) for all
(3)
are satisfied by more than one sequence.
Problem 9
Harold, Tanya, and Ulysses paint a very long picket fence. Harold starts with the first picket and paints every th picket; Tanya starts with the second picket and paints every
th picket; and Ulysses starts with the third picket and paints every
th picket. Call the positive integer
when the triple
of positive integers results in every picket being painted exactly once. Find the sum of all the paintable integers.
Problem 10
In the diagram below, angle is a right angle. Point
is on
, and
bisects angle
. Points
and
are on
and
, respectively, so that
and
. Given that
and
, find the integer closest to the area of quadrilateral
.
Problem 11
Let and
be two faces of a cube with
. A beam of light emanates from vertex
and reflects off face
at point
, which is
units from
and
units from
. The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point
until it next reaches a vertex of the cube is given by
, where
and
are integers and
is not divisible by the square of any prime. Find
.
Problem 12
Let for all complex numbers
, and let
for all positive integers
. Given that
and
, where
and
are real numbers, find
.
Problem 13
In triangle the medians
and
have lengths 18 and 27, respectively, and
. Extend
to intersect the circumcircle of
at
. The area of triangle
is
, where
and
are positive integers and
is not divisible by the square of any prime. Find
.
Problem 14
A set of distinct positive integers has the following property: for every integer
in
the arithmetic mean of the set of values obtained by deleting
from
is an integer. Given that 1 belongs to
and that 2002 is the largest element of
what is the greatest number of elements that
can have?
Problem 15
Polyhedron has six faces. Face
is a square with
face
is a trapezoid with
parallel to
and
and face
has
The other three faces are
and
The distance from
to face
is 12. Given that
where
and
are positive integers and
is not divisible by the square of any prime, find
See also
2002 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2001 AIME II Problems |
Followed by 2002 AIME II 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.