Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1998 AIME Problems"

m (→‎Problem 1)
m (Undo revision 205872 by Jinwoo (talk))
(Tag: Undo)
 
(12 intermediate revisions by 8 users not shown)
Line 1: Line 1:
 +
{{AIME Problems|year=1998}}
 +
 
== Problem 1 ==
 
== Problem 1 ==
 
For how many values of <math>k</math> is <math>12^{12}</math> the [[least common multiple]] of the positive integers <math>6^6</math> and <math>8^8</math>, and <math>k</math>?
 
For how many values of <math>k</math> is <math>12^{12}</math> the [[least common multiple]] of the positive integers <math>6^6</math> and <math>8^8</math>, and <math>k</math>?
Line 5: Line 7:
  
 
== Problem 2 ==
 
== Problem 2 ==
−
Find the number of [[ordered pair]]s <math>\displaystyle (x,y)</math> of positive integers that satisfy <math>x \le 2y \le 60 \displaystyle</math> and <math>\displaystyle y \le 2x \le 60</math>.
+
Find the number of [[ordered pair]]s <math>(x,y)</math> of positive integers that satisfy <math>x \le 2y \le 60</math> and <math>y \le 2x \le 60</math>.
  
 
[[1998 AIME Problems/Problem 2|Solution]]
 
[[1998 AIME Problems/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
−
The graph of <math> y^2 + 2xy + 40|x| \displaystyle = 400</math> partitions the plane into several regions.  What is the area of the bounded region?
+
The graph of <math> y^2 + 2xy + 40|x|= 400</math> partitions the plane into several regions.  What is the area of the bounded region?
  
 
[[1998 AIME Problems/Problem 3|Solution]]
 
[[1998 AIME Problems/Problem 3|Solution]]
Line 25: Line 27:
  
 
== Problem 6 ==
 
== Problem 6 ==
−
Let <math>A\displaystyle BCD</math> be a [[parallelogram]].  Extend <math>\overline{DA}</math> through <math>A</math> to a point <math>P,</math> and let <math>\overline{PC}</math> meet <math>\overline{AB}</math> at <math>Q</math> and <math>\overline{DB}</math> at <math>R.</math>  Given that <math>PQ = 735</math> and <math>QR = 112,</math> find <math>RC.</math>
+
Let <math>ABCD</math> be a [[parallelogram]].  Extend <math>\overline{DA}</math> through <math>A</math> to a point <math>P,</math> and let <math>\overline{PC}</math> meet <math>\overline{AB}</math> at <math>Q</math> and <math>\overline{DB}</math> at <math>R.</math>  Given that <math>PQ = 735</math> and <math>QR = 112,</math> find <math>RC.</math>
  
 
[[1998 AIME Problems/Problem 6|Solution]]
 
[[1998 AIME Problems/Problem 6|Solution]]
  
 
== Problem 7 ==
 
== Problem 7 ==
−
Let <math>n</math> be the number of ordered quadruples <math>\displaystyle(x_1,x_2,x_3,x_4)</math> of positive odd [[integer]]s that satisfy <math>\sum_{i = 1}^4 x_i = 98.</math>  Find <math>\frac n{100}.</math>
+
Let <math>n</math> be the number of ordered quadruples <math>(x_1,x_2,x_3,x_4)</math> of positive odd [[integer]]s that satisfy <math>\sum_{i = 1}^4 x_i = 98.</math>  Find <math>\frac n{100}.</math>
  
 
[[1998 AIME Problems/Problem 7|Solution]]
 
[[1998 AIME Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
−
Except for the first two terms, each term of the sequence <math>1000, x, 1000 - x,\ldots</math> is obtained by subtracting the preceding term from the one before that.  The last term of the sequence is the first [[negative]] term encounted.  What positive integer <math>x</math> produces a sequence of maximum length?
+
Except for the first two terms, each term of the sequence <math>1000, x, 1000 - x,\ldots</math> is obtained by subtracting the preceding term from the one before that.  The last term of the sequence is the first [[negative]] term encountered.  What positive integer <math>x</math> produces a sequence of maximum length?
  
 
[[1998 AIME Problems/Problem 8|Solution]]
 
[[1998 AIME Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
−
Two mathematicians take a morning coffee break each day.  They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly <math>m</math> mintues.  The [[probability]] that either one arrives while the other is in the cafeteria is <math>40 \%,</math> and <math>m = a - b\sqrt {c},</math> where <math>a, b,</math> and <math>c</math> are [[positive]] [[integer]]s, and <math>c</math> is not divisible by the square of any [[prime]].  Find <math>\displaystyle a + b + c.</math>
+
Two mathematicians take a morning coffee break each day.  They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly <math>m</math> minutes.  The [[probability]] that either one arrives while the other is in the cafeteria is <math>40 \%,</math> and <math>m = a - b\sqrt {c},</math> where <math>a, b,</math> and <math>c</math> are [[positive]] [[integer]]s, and <math>c</math> is not divisible by the square of any [[prime]].  Find <math>a + b + c.</math>
  
 
[[1998 AIME Problems/Problem 9|Solution]]
 
[[1998 AIME Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
−
Eight [[sphere]]s of [[radius]] 100 are placed on a flat [[plane|surface]] so that each sphere is [[tangent]] to two others and their [[center]]s are the vertices of a regular [[octagon]].  A ninth sphere is placed on the flat surface so that it is tangent to each of the other eight spheres.  The radius of this last sphere is <math>\displaystyle a + \displaystyle b\sqrt {c} \displaystyle,</math> where <math>a, b,</math> and <math>c</math> are [[positive]] [[integer]]s, and <math>c</math> is not divisible by the square of any [[prime]].  Find <math>\displaystyle a + b + c</math>.
+
Eight [[sphere]]s of [[radius]] 100 are placed on a flat [[plane|surface]] so that each sphere is [[tangent]] to two others and their [[center]]s are the vertices of a regular [[octagon]].  A ninth sphere is placed on the flat surface so that it is tangent to each of the other eight spheres.  The radius of this last sphere is <math>a +b\sqrt {c},</math> where <math>a, b,</math> and <math>c</math> are [[positive]] [[integer]]s, and <math>c</math> is not divisible by the square of any [[prime]].  Find <math>a + b + c</math>.
  
 
[[1998 AIME Problems/Problem 10|Solution]]
 
[[1998 AIME Problems/Problem 10|Solution]]
Line 55: Line 57:
  
 
== Problem 12 ==
 
== Problem 12 ==
−
Let <math>ABC</math> be [[equilateral triangle|equilateral]], and <math>D, E,</math> and <math>F</math> be the [[midpoint]]s of <math>\overline{BC}, \overline{CA},</math> and <math>\overline{AB},</math> respectively.  There exist [[point]]s <math>P, Q,</math> and <math>R</math> on <math>\displaystyle \overline{DE}, \overline{EF},</math> and <math>\overline{FD}, \displaystyle</math> respectively, with the property that <math>P</math> is on <math>\overline{CQ}, Q</math> is on <math>\overline{AR}, \displaystyle</math> and <math>R</math> is on <math>\overline{BP}.</math>  The [[ratio]] of the area of triangle <math>ABC</math> to the area of triangle <math>PQR</math> is <math>a + b\sqrt {c}, \displaystyle</math> where <math>a, b</math> and <math>c</math> are integers, and <math>c</math> is not divisible by the square of any [[prime]].  What is <math>a^{2} + b^{2} + c^{2}</math>?
+
Let <math>ABC</math> be [[equilateral triangle|equilateral]], and <math>D, E,</math> and <math>F</math> be the [[midpoint]]s of <math>\overline{BC}, \overline{CA},</math> and <math>\overline{AB},</math> respectively.  There exist [[point]]s <math>P, Q,</math> and <math>R</math> on <math>\overline{DE}, \overline{EF},</math> and <math>\overline{FD},</math> respectively, with the property that <math>P</math> is on <math>\overline{CQ}, Q</math> is on <math>\overline{AR},</math> and <math>R</math> is on <math>\overline{BP}.</math>  The [[ratio]] of the area of triangle <math>ABC</math> to the area of triangle <math>PQR</math> is <math>a + b\sqrt {c},</math> where <math>a, b</math> and <math>c</math> are integers, and <math>c</math> is not divisible by the square of any [[prime]].  What is <math>a^{2} + b^{2} + c^{2}</math>?
−
 
 
−
[[Image:1998_AIME-12.png]]
 
  
 
[[1998 AIME Problems/Problem 12|Solution]]
 
[[1998 AIME Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
−
If <math>\{a_1,a_2,a_3,\ldots,a_n\}</math> is a [[set]] of [[real numbers]], indexed so that <math>\displaystyle a_1 < a_2 < a_3 < \displaystyle  \cdots < a_n,</math> its complex power sum is defined to be <math>\displaystyle a_1i + a_2i^2 \displaystyle + a_3i^3 + \cdots + a_ni^n,</math> where <math>i^2 = - 1.</math>  Let <math>S_n</math> be the sum of the complex power sums of all nonempty [[subset]]s of <math>\displaystyle \{1,2,\ldots,n\}.</math>  Given that <math>S_8 = - 176 - 64i</math> and <math>\displaystyle  S_9 = p + qi,</math> were <math>p</math> and <math>q</math> are integers, find <math>|p| + |q|.</math>
+
If <math>\{a_1,a_2,a_3,\ldots,a_n\}</math> is a [[set]] of [[real numbers]], indexed so that <math>a_1 < a_2 < a_3 < \cdots < a_n,</math> its complex power sum is defined to be <math>a_1i + a_2i^2+ a_3i^3 + \cdots + a_ni^n,</math> where <math>i^2 = - 1.</math>  Let <math>S_n</math> be the sum of the complex power sums of all nonempty [[subset]]s of <math>\{1,2,\ldots,n\}.</math>  Given that <math>S_8 = - 176 - 64i</math> and <math> S_9 = p + qi,</math> where <math>p</math> and <math>q</math> are integers, find <math>|p| + |q|.</math>
  
 
[[1998 AIME Problems/Problem 13|Solution]]
 
[[1998 AIME Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
−
An <math>m\times n\times p</math> rectangular box has half the volume of an <math>\displaystyle (m + 2)\times(n + 2)\times(p + 2)</math> rectangular box, where <math>m, n,</math> and <math>p</math> are integers, and <math>m\le n\le p.</math>  What is the largest possible value of <math>p</math>?
+
An <math>m\times n\times p</math> rectangular box has half the volume of an <math>(m + 2)\times(n + 2)\times(p + 2)</math> rectangular box, where <math>m, n,</math> and <math>p</math> are integers, and <math>m\le n\le p.</math>  What is the largest possible value of <math>p</math>?
  
 
[[1998 AIME Problems/Problem 14|Solution]]
 
[[1998 AIME Problems/Problem 14|Solution]]
Line 81: Line 81:
 
*[[Mathematics competition resources]]
 
*[[Mathematics competition resources]]
  
−
{{AIME box|year = 1998|before=[[1997 AIME]]|after=[[1999 AIME]]}}
+
{{AIME box|year = 1998|before=[[1997 AIME Problems]]|after=[[1999 AIME Problems]]}}
 +
{{MAA Notice}}

Latest revision as of 01:01, 28 November 2023

1998 AIME (Answer Key)
Printable version | AoPS Contest Collections • PDF

Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Problem 1

For how many values of $k$ is $12^{12}$ the least common multiple of the positive integers $6^6$ and $8^8$, and $k$?

Solution

Problem 2

Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x \le 2y \le 60$ and $y \le 2x \le 60$.

Solution

Problem 3

The graph of $y^2 + 2xy + 40|x|= 400$ partitions the plane into several regions. What is the area of the bounded region?

Solution

Problem 4

Nine tiles are numbered $1, 2, 3, \cdots, 9,$ respectively. Each of three players randomly selects and keeps three of the tiles, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 5

Given that $A_k = \frac {k(k - 1)}2\cos\frac {k(k - 1)\pi}2,$ find $|A_{19} + A_{20} + \cdots + A_{98}|.$

Solution

Problem 6

Let $ABCD$ be a parallelogram. Extend $\overline{DA}$ through $A$ to a point $P,$ and let $\overline{PC}$ meet $\overline{AB}$ at $Q$ and $\overline{DB}$ at $R.$ Given that $PQ = 735$ and $QR = 112,$ find $RC.$

Solution

Problem 7

Let $n$ be the number of ordered quadruples $(x_1,x_2,x_3,x_4)$ of positive odd integers that satisfy $\sum_{i = 1}^4 x_i = 98.$ Find $\frac n{100}.$

Solution

Problem 8

Except for the first two terms, each term of the sequence $1000, x, 1000 - x,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $x$ produces a sequence of maximum length?

Solution

Problem 9

Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40 \%,$ and $m = a - b\sqrt {c},$ where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a + b + c.$

Solution

Problem 10

Eight spheres of radius 100 are placed on a flat surface so that each sphere is tangent to two others and their centers are the vertices of a regular octagon. A ninth sphere is placed on the flat surface so that it is tangent to each of the other eight spheres. The radius of this last sphere is $a +b\sqrt {c},$ where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a + b + c$.

Solution

Problem 11

Three of the edges of a cube are $\overline{AB}, \overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\overline{AB}, \overline{BC},$ and $\overline{CD},$ respectively, so that $AP = 5, PB = 15, BQ = 15,$ and $CR = 10.$ What is the area of the polygon that is the intersection of plane $PQR$ and the cube?

Solution

Problem 12

Let $ABC$ be equilateral, and $D, E,$ and $F$ be the midpoints of $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively. There exist points $P, Q,$ and $R$ on $\overline{DE}, \overline{EF},$ and $\overline{FD},$ respectively, with the property that $P$ is on $\overline{CQ}, Q$ is on $\overline{AR},$ and $R$ is on $\overline{BP}.$ The ratio of the area of triangle $ABC$ to the area of triangle $PQR$ is $a + b\sqrt {c},$ where $a, b$ and $c$ are integers, and $c$ is not divisible by the square of any prime. What is $a^{2} + b^{2} + c^{2}$?

Solution

Problem 13

If $\{a_1,a_2,a_3,\ldots,a_n\}$ is a set of real numbers, indexed so that $a_1 < a_2 < a_3 < \cdots < a_n,$ its complex power sum is defined to be $a_1i + a_2i^2+ a_3i^3 + \cdots + a_ni^n,$ where $i^2 = - 1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\{1,2,\ldots,n\}.$ Given that $S_8 = - 176 - 64i$ and $S_9 = p + qi,$ where $p$ and $q$ are integers, find $|p| + |q|.$

Solution

Problem 14

An $m\times n\times p$ rectangular box has half the volume of an $(m + 2)\times(n + 2)\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\le n\le p.$ What is the largest possible value of $p$?

Solution

Problem 15

Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i,j)$ and $(j,i)$ do not both appear for any $i$ and $j$. Let $D_{40}$ be the set of all dominos whose coordinates are no larger than 40. Find the length of the longest proper sequence of dominos that can be formed using the dominos of $D_{40}.$

Solution

See also

1998 AIME (Problems • Answer Key • Resources)
Preceded by
1997 AIME Problems 		Followed by
1999 AIME Problems
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1998_AIME_Problems&oldid=206206"