Difference between revisions of "1998 AIME Problems"
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== 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>\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>? | 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>? | ||
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[[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>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> | + | 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]] | ||
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*[[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}} | {{MAA Notice}} |
Latest revision as of 02:01, 28 November 2023
1998 AIME (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
For how many values of is
the least common multiple of the positive integers
and
, and
?
Problem 2
Find the number of ordered pairs of positive integers that satisfy
and
.
Problem 3
The graph of partitions the plane into several regions. What is the area of the bounded region?
Problem 4
Nine tiles are numbered 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
where
and
are relatively prime positive integers. Find
Problem 5
Given that find
Problem 6
Let be a parallelogram. Extend
through
to a point
and let
meet
at
and
at
Given that
and
find
Problem 7
Let be the number of ordered quadruples
of positive odd integers that satisfy
Find
Problem 8
Except for the first two terms, each term of the sequence 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
produces a sequence of maximum length?
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 minutes. The probability that either one arrives while the other is in the cafeteria is
and
where
and
are positive integers, and
is not divisible by the square of any prime. Find
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 where
and
are positive integers, and
is not divisible by the square of any prime. Find
.
Problem 11
Three of the edges of a cube are and
and
is an interior diagonal. Points
and
are on
and
respectively, so that
and
What is the area of the polygon that is the intersection of plane
and the cube?
Problem 12
Let be equilateral, and
and
be the midpoints of
and
respectively. There exist points
and
on
and
respectively, with the property that
is on
is on
and
is on
The ratio of the area of triangle
to the area of triangle
is
where
and
are integers, and
is not divisible by the square of any prime. What is
?
Problem 13
If is a set of real numbers, indexed so that
its complex power sum is defined to be
where
Let
be the sum of the complex power sums of all nonempty subsets of
Given that
and
where
and
are integers, find
Problem 14
An rectangular box has half the volume of an
rectangular box, where
and
are integers, and
What is the largest possible value of
?
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 and
do not both appear for any
and
. Let
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
See also
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
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.