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Difference between revisions of "1998 AIME Problems"

 
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== Problem 1 ==
 
== Problem 1 ==
 +
For how many values of <math>\displaystyle k</math> is <math>\displaystyle 12^{12}</math> the [[least common multiple]] of the positive integers <math>6^6</math> and <math>8^8</math>?
  
 
[[1998 AIME Problems/Problem 1|Solution]]
 
[[1998 AIME Problems/Problem 1|Solution]]
  
 
== 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>.
  
 
[[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?
  
 
[[1998 AIME Problems/Problem 3|Solution]]
 
[[1998 AIME Problems/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== 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]]
 
[[1998 AIME Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== 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]]
 
[[1998 AIME Problems/Problem 5|Solution]]
  
 
== 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>
  
 
[[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>
  
 
[[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?
  
 
[[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>
  
 
[[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>.
  
 
[[1998 AIME Problems/Problem 10|Solution]]
 
[[1998 AIME Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== 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]]
 
[[1998 AIME Problems/Problem 11|Solution]]
  
 
== 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>?
 +
 +
[[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>
  
 
[[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>?
  
 
[[1998 AIME Problems/Problem 14|Solution]]
 
[[1998 AIME Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== 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]]
 
[[1998 AIME Problems/Problem 15|Solution]]
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* [[American Invitational Mathematics Examination]]
 
* [[American Invitational Mathematics Examination]]
 
* [[AIME Problems and Solutions]]
 
* [[AIME Problems and Solutions]]
−
* [[Mathematics competition resources]]
+
*[[Mathematics competition resources]]
 +
 
 +
{{AIME box|year = 1998|before=[[1997 AIME]]|after=[[1999 AIME]]}}

Revision as of 19:12, 9 September 2007

Problem 1

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

Solution

Problem 2

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

Solution

Problem 3

The graph of $y^2 + 2xy + 40|x| \displaystyle = 400$ 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 $A\displaystyle BCD$ 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 $\displaystyle(x_1,x_2,x_3,x_4)$ 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 encounted. 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 $m$ mintues. 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 $\displaystyle a + b + c.$

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 $\displaystyle a + \displaystyle b\sqrt {c} \displaystyle,$ where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $\displaystyle a + b + c$.

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 $\displaystyle \overline{DE}, \overline{EF},$ and $\overline{FD}, \displaystyle$ respectively, with the property that $P$ is on $\overline{CQ}, Q$ is on $\overline{AR}, \displaystyle$ and $R$ is on $\overline{BP}.$ The ratio of the area of triangle $ABC$ to the area of triangle $PQR$ is $a + b\sqrt {c}, \displaystyle$ where $a, b$ and $c$ are integers, and $c$ is not divisible by the square of any prime. What is $a^{2} + b^{2} + c^{2}$?

1998 AIME-12.png

Solution

Problem 13

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

Solution

Problem 14

An $m\times n\times p$ rectangular box has half the volume of an $\displaystyle (m + 2)\times(n + 2)\times(p + 2)$ 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 		Followed by
1999 AIME
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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