Difference between revisions of "1998 AIME Problems"

Latest revision as of 01:01, 28 November 2023

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

@@ Line 1: / Line 1: @@
+{{AIME Problems|year=1998}}
 == 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>?
 [[1998 AIME Problems/Problem 1|Solution]]
 == Problem 2 ==
+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]]
 == Problem 3 ==
+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]]
 == Problem 4 ==
+Nine tiles are numbered <math>1, 2, 3, \cdots, 9,</math> 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 <math>m/n,</math> where <math>m</math> and <math>n</math> are [[relatively prime]] [[positive integer]]s.  Find <math>m+n.</math>
 [[1998 AIME Problems/Problem 4|Solution]]
 == Problem 5 ==
+Given that <math>A_k = \frac {k(k - 1)}2\cos\frac {k(k - 1)\pi}2,</math> find <math>|A_{19} + A_{20} + \cdots + A_{98}|.</math>
 [[1998 AIME Problems/Problem 5|Solution]]
 == Problem 6 ==
+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]]
 == Problem 7 ==
+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]]
 == 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 encountered.  What positive integer <math>x</math> produces a sequence of maximum length?
 [[1998 AIME Problems/Problem 8|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 <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]]
 == 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>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]]
 == Problem 11 ==
+Three of the edges of a [[cube]] are <math>\overline{AB}, \overline{BC},</math> and <math>\overline{CD},</math> and <math>\overline{AD}</math> is an interior [[diagonal]].  Points <math>P, Q,</math> and <math>R</math> are on <math>\overline{AB}, \overline{BC},</math> and <math>\overline{CD},</math> respectively, so that <math>AP = 5, PB = 15, BQ = 15,</math> and <math>CR = 10.</math>  What is the [[area]] of the [[polygon]] that is the [[intersection]] of [[plane]] <math>PQR</math> and the cube?
 [[1998 AIME Problems/Problem 11|Solution]]
 == 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>\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>?
 [[1998 AIME Problems/Problem 12|Solution]]
 == Problem 13 ==
+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]]
 == Problem 14 ==
+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]]
 == 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 <math>(i,j)</math> and <math>(j,i)</math> do not both appear for any <math>i</math> and <math>j</math>.  Let <math>D_{40}</math> 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 <math>D_{40}.</math>
 [[1998 AIME Problems/Problem 15|Solution]]
@@ Line 63: / Line 79: @@
 * [[American Invitational Mathematics Examination]]
 * [[AIME Problems and Solutions]]
-* [[Mathematics competition resources]]
+*[[Mathematics competition resources]]
+{{AIME box|year = 1998|before=[[1997 AIME Problems]]|after=[[1999 AIME Problems]]}}
+{{MAA Notice}}

1998 AIME (Problems • Answer Key • Resources)
Preceded by 1997 AIME Problems	Followed by 1999 AIME Problems
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Difference between revisions of "1998 AIME Problems"

Latest revision as of 01:01, 28 November 2023

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

See also