Difference between revisions of "2002 AIME I Problems"
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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 number 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? | + | 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 number 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]] | ||
Line 31: | Line 39: | ||
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 65: | Line 77: | ||
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]] | ||
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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 21:12, 19 February 2019
2002 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
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 number 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 ,
What are the first three digits to the right of the decimal point in the decimal representation of ?
Problem 8
Find the smallest integer k 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.