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Difference between revisions of "1998 AIME Problems"

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== Problem 1 ==
 
== Problem 1 ==
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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>?
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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]]
 
[[1998 AIME Problems/Problem 1|Solution]]

Revision as of 15:16, 17 March 2008

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 $\displaystyle (x,y)$ 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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