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Difference between revisions of "1987 AIME Problems"

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== Problem 1 ==
 
== Problem 1 ==
 +
An ordered pair <math>\displaystyle (m,n)</math> of non-negative integers is called "simple" if the addition <math>\displaystyle m+n</math> in base <math>\displaystyle 10</math> requires no carrying. Find the number of simple ordered pairs of non-negative integers that sum to <math>\displaystyle 1492</math>.
  
 
[[1987 AIME Problems/Problem 1|Solution]]
 
[[1987 AIME Problems/Problem 1|Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
 +
What is the largest possible distance between two points, one on the sphere of radius 19 with center <math>\displaystyle (-2,-10,-5)</math> and the other on the sphere of radius 87 with center <math>\displaystyle (12,8,-16)</math>?
  
 
[[1987 AIME Problems/Problem 2|Solution]]
 
[[1987 AIME Problems/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
 +
By a proper divisior of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?
  
 
[[1987 AIME Problems/Problem 3|Solution]]
 
[[1987 AIME Problems/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
 +
Find the area of the region enclosed by the graph of <math>\displaystyle |x-60|+|y|=|x/4|.</math>
  
 
[[1987 AIME Problems/Problem 4|Solution]]
 
[[1987 AIME Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
 +
Find <math>\displaystyle 3x^2 y^2</math> if <math>\displaystyle x</math> and <math>\displaystyle y</math> are integers such that <math>\displaystyle y^2 + 3x^2 y^2 = 30x^2 + 517</math>.
  
 
[[1987 AIME Problems/Problem 5|Solution]]
 
[[1987 AIME Problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
 +
Rectangle <math>\displaystyle ABCD</math> is divided into four parts of equal area by five segments as shown in the figure, where <math>\displaystyle XY = YB + BC + CZ = ZW = WD + DA + AX</math>, and <math>\displaystyle PQ</math> is parallel to <math>\displaystyle AB</math>.  Find the length of <math>\displaystyle AB</math> (in cm) if <math>\displaystyle BC = 19</math> cm and <math>\displaystyle PQ = 87</math> cm.
 +
 +
[[Image:AIME_1987_Problem_6.png]]
  
 
[[1987 AIME Problems/Problem 6|Solution]]
 
[[1987 AIME Problems/Problem 6|Solution]]
  
 
== Problem 7 ==
 
== Problem 7 ==
−
 
+
Let <math>\displaystyle [r,s]</math> denote the least common multiple of positive integers <math>\displaystyle r</math> and <math>\displaystyle s</math>.  Find the number of ordered triples <math>\displaystyle (a,b,c)</math> of positive integers for which <math>\displaystyle [a,b] = 1000</math>, <math>\displaystyle [b,c] = 2000</math>, and <math>\displaystyle [c,a] = 2000</math>.
  
 
[[1987 AIME Problems/Problem 7|Solution]]
 
[[1987 AIME Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
 +
What is the largest positive integer <math>\displaystyle n</math> for which there is a unique integer <math>\displaystyle k</math> such that <math>\displaystyle \frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}</math>?
  
 
[[1987 AIME Problems/Problem 8|Solution]]
 
[[1987 AIME Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
 +
Triangle <math>\displaystyle ABC</math> has right angle at <math>\displaystyle B</math>, and contains a point <math>\displaystyle P</math> for which <math>\displaystyle PA = 10</math>, <math>\displaystyle PB = 6</math>, and <math>\displaystyle \angle APB = \angle BPC = \angle CPA</math>.  Find <math>\displaystyle PC</math>.
 +
 +
[[Image:AIME_1987_Problem_9.png]]
  
 
[[1987 AIME Problems/Problem 9|Solution]]
 
[[1987 AIME Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
 +
Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is constant.)
  
 
[[1987 AIME Problems/Problem 10|Solution]]
 
[[1987 AIME Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== Problem 11 ==
 +
Find the largest possible value of <math>\displaystyle k</math> for which <math>\displaystyle 3^{11}</math> is expressible as the sum of <math>\displaystyle k</math> consecutive positive integers.
  
 
[[1987 AIME Problems/Problem 11|Solution]]
 
[[1987 AIME Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
 +
Let <math>\displaystyle m</math> be the smallest integer whose cube root is of the form <math>\displaystyle n+r</math>, where <math>\displaystyle n</math> is a positive integer and <math>\displaystyle r</math> is a positive real number less than <math>\displaystyle 1/1000</math>. Find <math>\displaystyle n</math>.
  
 
[[1987 AIME Problems/Problem 12|Solution]]
 
[[1987 AIME Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
 +
A given sequence <math>\displaystyle r_1, r_2, \dots, r_n</math> of distinct real numbers can be put in ascending order by means of one or more "bubble passes".  A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller, then comparing the third term with the second term and exchanging them if and only if the third term is smaller, and so on in order, through comparing the last term, <math>\displaystyle r_n</math>, with its current predecessor and exchanging them if and only if the last term is smaller.
 +
 +
The example below shows how the sequence 1, 9, 8, 7 is transformed into the sequence 1, 8, 7, 9 by one bubble pass.  The numbers compared at each step are underlined.
 +
<center><math>\underline{1 \quad 9} \quad 8 \quad 7</math></center>
 +
<center><math>1 \quad {}\underline{9 \quad 8} \quad 7</math></center>
 +
<center><math>1 \quad 8 \quad \underline{9 \quad 7}</math></center>
 +
<center><math>1 \quad 8 \quad 7 \quad 9</math></center>
 +
Suppose that <math>\displaystyle n = 40</math>, and that the terms of the initial sequence <math>\displaystyle r_1, r_2, \dots, r_{40}</math> are distinct from one another and are in random order.  Let <math>\displaystyle p/q</math>, in lowest terms, be the probability that the number that begins as <math>\displaystyle r_{20}</math> will end up, after one bubble pass, in the <math>\displaystyle 30^{\mbox{th}}</math> place.  Find <math>\displaystyle p + q</math>.
  
 
[[1987 AIME Problems/Problem 13|Solution]]
 
[[1987 AIME Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
 +
Compute
 +
<center><math>\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}</math></center>.
  
 
[[1987 AIME Problems/Problem 14|Solution]]
 
[[1987 AIME Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==
 +
Squares <math>\displaystyle S_1</math> and <math>\displaystyle S_2</math> are inscribed in right triangle <math>\displaystyle ABC</math>, as shown in the figures below. Find <math>\displaystyle AC + CB</math> if area <math>\displaystyle (S_1) = 441</math> and area <math>\displaystyle (S_2) = 440</math>.
 +
 +
[[Image:AIME_1987_Problem_15.png]]
  
 
[[1987 AIME Problems/Problem 15|Solution]]
 
[[1987 AIME Problems/Problem 15|Solution]]

Revision as of 00:14, 11 February 2007

Problem 1

An ordered pair $\displaystyle (m,n)$ of non-negative integers is called "simple" if the addition $\displaystyle m+n$ in base $\displaystyle 10$ requires no carrying. Find the number of simple ordered pairs of non-negative integers that sum to $\displaystyle 1492$.

Solution

Problem 2

What is the largest possible distance between two points, one on the sphere of radius 19 with center $\displaystyle (-2,-10,-5)$ and the other on the sphere of radius 87 with center $\displaystyle (12,8,-16)$?

Solution

Problem 3

By a proper divisior of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?

Solution

Problem 4

Find the area of the region enclosed by the graph of $\displaystyle |x-60|+|y|=|x/4|.$

Solution

Problem 5

Find $\displaystyle 3x^2 y^2$ if $\displaystyle x$ and $\displaystyle y$ are integers such that $\displaystyle y^2 + 3x^2 y^2 = 30x^2 + 517$.

Solution

Problem 6

Rectangle $\displaystyle ABCD$ is divided into four parts of equal area by five segments as shown in the figure, where $\displaystyle XY = YB + BC + CZ = ZW = WD + DA + AX$, and $\displaystyle PQ$ is parallel to $\displaystyle AB$. Find the length of $\displaystyle AB$ (in cm) if $\displaystyle BC = 19$ cm and $\displaystyle PQ = 87$ cm.

AIME 1987 Problem 6.png

Solution

Problem 7

Let $\displaystyle [r,s]$ denote the least common multiple of positive integers $\displaystyle r$ and $\displaystyle s$. Find the number of ordered triples $\displaystyle (a,b,c)$ of positive integers for which $\displaystyle [a,b] = 1000$, $\displaystyle [b,c] = 2000$, and $\displaystyle [c,a] = 2000$.

Solution

Problem 8

What is the largest positive integer $\displaystyle n$ for which there is a unique integer $\displaystyle k$ such that $\displaystyle \frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}$?

Solution

Problem 9

Triangle $\displaystyle ABC$ has right angle at $\displaystyle B$, and contains a point $\displaystyle P$ for which $\displaystyle PA = 10$, $\displaystyle PB = 6$, and $\displaystyle \angle APB = \angle BPC = \angle CPA$. Find $\displaystyle PC$.

AIME 1987 Problem 9.png

Solution

Problem 10

Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is constant.)

Solution

Problem 11

Find the largest possible value of $\displaystyle k$ for which $\displaystyle 3^{11}$ is expressible as the sum of $\displaystyle k$ consecutive positive integers.

Solution

Problem 12

Let $\displaystyle m$ be the smallest integer whose cube root is of the form $\displaystyle n+r$, where $\displaystyle n$ is a positive integer and $\displaystyle r$ is a positive real number less than $\displaystyle 1/1000$. Find $\displaystyle n$.

Solution

Problem 13

A given sequence $\displaystyle r_1, r_2, \dots, r_n$ of distinct real numbers can be put in ascending order by means of one or more "bubble passes". A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller, then comparing the third term with the second term and exchanging them if and only if the third term is smaller, and so on in order, through comparing the last term, $\displaystyle r_n$, with its current predecessor and exchanging them if and only if the last term is smaller.

The example below shows how the sequence 1, 9, 8, 7 is transformed into the sequence 1, 8, 7, 9 by one bubble pass. The numbers compared at each step are underlined.

$\underline{1 \quad 9} \quad 8 \quad 7$
$1 \quad {}\underline{9 \quad 8} \quad 7$
$1 \quad 8 \quad \underline{9 \quad 7}$
$1 \quad 8 \quad 7 \quad 9$

Suppose that $\displaystyle n = 40$, and that the terms of the initial sequence $\displaystyle r_1, r_2, \dots, r_{40}$ are distinct from one another and are in random order. Let $\displaystyle p/q$, in lowest terms, be the probability that the number that begins as $\displaystyle r_{20}$ will end up, after one bubble pass, in the $\displaystyle 30^{\mbox{th}}$ place. Find $\displaystyle p + q$.

Solution

Problem 14

Compute

$\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}$

.

Solution

Problem 15

Squares $\displaystyle S_1$ and $\displaystyle S_2$ are inscribed in right triangle $\displaystyle ABC$, as shown in the figures below. Find $\displaystyle AC + CB$ if area $\displaystyle (S_1) = 441$ and area $\displaystyle (S_2) = 440$.

AIME 1987 Problem 15.png

Solution

See also

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