Difference between revisions of "1988 AIME Problems"

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{{AIME Problems|year=1988}}
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== Problem 1 ==
 
== Problem 1 ==
 
One commercially available ten-button lock may be opened by depressing -- in any order -- the correct five buttons. The sample shown below has <math>\{1, 2, 3, 6, 9\}</math> as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow?
 
One commercially available ten-button lock may be opened by depressing -- in any order -- the correct five buttons. The sample shown below has <math>\{1, 2, 3, 6, 9\}</math> as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow?
[[Image:1988-1.png]]
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<center>[[Image:1988-1.png]]</center>
  
 
[[1988 AIME Problems/Problem 1|Solution]]
 
[[1988 AIME Problems/Problem 1|Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
 +
For any positive integer <math>k</math>, let <math>f_1(k)</math> denote the square of the sum of the digits of <math>k</math>.  For <math>n \ge 2</math>, let <math>f_n(k) = f_1(f_{n - 1}(k))</math>.  Find <math>f_{1988}(11)</math>.
  
 
[[1988 AIME Problems/Problem 2|Solution]]
 
[[1988 AIME Problems/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
 +
Find <math>(\log_2 x)^2</math> if <math>\log_2 (\log_8 x) = \log_8 (\log_2 x)</math>.
  
 
[[1988 AIME Problems/Problem 3|Solution]]
 
[[1988 AIME Problems/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
 +
Suppose that <math>|x_i| < 1</math> for <math>i = 1, 2, \dots, n</math>. Suppose further that
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<center><math>|x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|</math>.</center>
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What is the smallest possible value of <math>n</math>?
  
 
[[1988 AIME Problems/Problem 4|Solution]]
 
[[1988 AIME Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
 +
Let <math>\frac{m}{n}</math>, in lowest terms, be the probability that a randomly chosen positive divisor of <math>10^{99}</math> is an integer multiple of <math>10^{88}</math>. Find <math>m + n</math>.
  
 
[[1988 AIME Problems/Problem 5|Solution]]
 
[[1988 AIME Problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
 +
It is possible to place positive integers into the vacant twenty-one squares of the 5 times 5 square shown below so that the numbers in each row and column form arithmetic sequences. Find the number that must occupy the vacant square marked by the asterisk (*).
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 +
<center>[[Image:AIME_1988_Problem_06.png]]</center>
  
 
[[1988 AIME Problems/Problem 6|Solution]]
 
[[1988 AIME Problems/Problem 6|Solution]]
  
 
== Problem 7 ==
 
== Problem 7 ==
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In triangle <math>ABC</math>, <math>\tan \angle CAB = 22/7</math>, and the altitude from <math>A</math> divides <math>BC</math> into segments of length <math>3</math> and <math>17</math>. What is the area of triangle <math>ABC</math>?
  
 
[[1988 AIME Problems/Problem 7|Solution]]
 
[[1988 AIME Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
 +
The function <math>f</math>, defined on the set of ordered pairs of positive integers, satisfies the following properties:
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<cmath>
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\begin{align*}
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f(x,x) &=x, \\
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f(x,y) &=f(y,x), \quad \text{and} \\
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(x + y) f(x,y) &= yf(x,x + y).
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\end{align*}
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</cmath>
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Calculate <math>f(14,52)</math>.
  
 
[[1988 AIME Problems/Problem 8|Solution]]
 
[[1988 AIME Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
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Find the smallest positive integer whose [[perfect cube|cube]] ends in <math>888</math>.
  
 
[[1988 AIME Problems/Problem 9|Solution]]
 
[[1988 AIME Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
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A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face?
  
 
[[1988 AIME Problems/Problem 10|Solution]]
 
[[1988 AIME Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== Problem 11 ==
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Let <math>w_1, w_2, \dots, w_n</math> be complex numbers. A line <math>L</math> in the complex plane is called a mean line for the points <math>w_1, w_2, \dots, w_n</math> if <math>L</math> contains points (complex numbers) <math>z_1, z_2, \dots, z_n</math> such that
 +
<center><math>\sum_{k = 1}^n (z_k - w_k) = 0.</math></center>
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For the numbers <math>w_1 = 32 + 170i</math>, <math>w_2 = -7 + 64i</math>, <math>w_3 = -9 +200i</math>, <math>w_4 = 1 + 27i</math>, and <math>w_5 = -14 + 43i</math>, there is a unique mean line with y-intercept <math>3</math>. Find the slope of this mean line.
  
 
[[1988 AIME Problems/Problem 11|Solution]]
 
[[1988 AIME Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
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Let <math>P</math> be an interior point of triangle <math>ABC</math> and extend lines from the vertices through <math>P</math> to the opposite sides. Let <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> denote the lengths of the segments indicated in the figure. Find the product <math>abc</math> if <math>a + b + c = 43</math> and <math>d = 3</math>.
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 +
<center>[[Image:AIME_1988_Problem_12.png]]</center>
  
 
[[1988 AIME Problems/Problem 12|Solution]]
 
[[1988 AIME Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
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Find <math>a</math> if <math>a</math> and <math>b</math> are integers such that <math>x^2 - x - 1</math> is a factor of <math>ax^{17} + bx^{16} + 1</math>.
  
 
[[1988 AIME Problems/Problem 13|Solution]]
 
[[1988 AIME Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
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Let <math>C</math> be the graph of <math>xy = 1</math>, and denote by <math>C^*</math> the reflection of <math>C</math> in the line <math>y = 2x</math>. Let the equation of <math>C^*</math> be written in the form
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<cmath>12x^2 + bxy + cy^2 + d = 0.</cmath>
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Find the product <math>bc</math>.
  
 
[[1988 AIME Problems/Problem 14|Solution]]
 
[[1988 AIME Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==
 +
In an office at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's in-box. When there is time, the secretary takes the top letter off the pile and types it. There are nine letters to be typed during the day, and the boss delivers them in the order 1, 2, 3, 4, 5, 6, 7, 8, 9.
 +
 +
While leaving for lunch, the secretary tells a colleague that letter 8 has already been typed, but says nothing else about the morning's typing. The colleague wonders which of the nine letters remain to be typed after lunch and in what order they will be typed. Based upon the above information, how many such after-lunch typing orders are possible? (That there are no letters left to be typed is one of the possibilities.)
  
 
[[1988 AIME Problems/Problem 15|Solution]]
 
[[1988 AIME Problems/Problem 15|Solution]]
  
 
== See also ==
 
== See also ==
 +
 +
{{AIME box|year=1988|before=[[1987 AIME Problems]]|after=[[1989 AIME Problems]]}}
 +
 
* [[American Invitational Mathematics Examination]]
 
* [[American Invitational Mathematics Examination]]
 
* [[AIME Problems and Solutions]]
 
* [[AIME Problems and Solutions]]
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[[Category:AIME Problems|1988]]
 
[[Category:AIME Problems|1988]]
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{{MAA Notice}}

Latest revision as of 06:35, 7 September 2018

1988 AIME (Answer Key)
Printable version | AoPS Contest Collections

Instructions

  1. 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.
  2. 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

One commercially available ten-button lock may be opened by depressing -- in any order -- the correct five buttons. The sample shown below has $\{1, 2, 3, 6, 9\}$ as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow?

1988-1.png

Solution

Problem 2

For any positive integer $k$, let $f_1(k)$ denote the square of the sum of the digits of $k$. For $n \ge 2$, let $f_n(k) = f_1(f_{n - 1}(k))$. Find $f_{1988}(11)$.

Solution

Problem 3

Find $(\log_2 x)^2$ if $\log_2 (\log_8 x) = \log_8 (\log_2 x)$.

Solution

Problem 4

Suppose that $|x_i| < 1$ for $i = 1, 2, \dots, n$. Suppose further that

$|x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|$.

What is the smallest possible value of $n$?

Solution

Problem 5

Let $\frac{m}{n}$, in lowest terms, be the probability that a randomly chosen positive divisor of $10^{99}$ is an integer multiple of $10^{88}$. Find $m + n$.

Solution

Problem 6

It is possible to place positive integers into the vacant twenty-one squares of the 5 times 5 square shown below so that the numbers in each row and column form arithmetic sequences. Find the number that must occupy the vacant square marked by the asterisk (*).

AIME 1988 Problem 06.png

Solution

Problem 7

In triangle $ABC$, $\tan \angle CAB = 22/7$, and the altitude from $A$ divides $BC$ into segments of length $3$ and $17$. What is the area of triangle $ABC$?

Solution

Problem 8

The function $f$, defined on the set of ordered pairs of positive integers, satisfies the following properties: \begin{align*}  f(x,x) &=x, \\  f(x,y) &=f(y,x), \quad \text{and} \\ (x + y) f(x,y) &= yf(x,x + y).  \end{align*} Calculate $f(14,52)$.

Solution

Problem 9

Find the smallest positive integer whose cube ends in $888$.

Solution

Problem 10

A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face?

Solution

Problem 11

Let $w_1, w_2, \dots, w_n$ be complex numbers. A line $L$ in the complex plane is called a mean line for the points $w_1, w_2, \dots, w_n$ if $L$ contains points (complex numbers) $z_1, z_2, \dots, z_n$ such that

$\sum_{k = 1}^n (z_k - w_k) = 0.$

For the numbers $w_1 = 32 + 170i$, $w_2 = -7 + 64i$, $w_3 = -9 +200i$, $w_4 = 1 + 27i$, and $w_5 = -14 + 43i$, there is a unique mean line with y-intercept $3$. Find the slope of this mean line.

Solution

Problem 12

Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$.

AIME 1988 Problem 12.png

Solution

Problem 13

Find $a$ if $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^{17} + bx^{16} + 1$.

Solution

Problem 14

Let $C$ be the graph of $xy = 1$, and denote by $C^*$ the reflection of $C$ in the line $y = 2x$. Let the equation of $C^*$ be written in the form \[12x^2 + bxy + cy^2 + d = 0.\] Find the product $bc$.

Solution

Problem 15

In an office at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's in-box. When there is time, the secretary takes the top letter off the pile and types it. There are nine letters to be typed during the day, and the boss delivers them in the order 1, 2, 3, 4, 5, 6, 7, 8, 9.

While leaving for lunch, the secretary tells a colleague that letter 8 has already been typed, but says nothing else about the morning's typing. The colleague wonders which of the nine letters remain to be typed after lunch and in what order they will be typed. Based upon the above information, how many such after-lunch typing orders are possible? (That there are no letters left to be typed is one of the possibilities.)

Solution

See also

1988 AIME (ProblemsAnswer KeyResources)
Preceded by
1987 AIME Problems
Followed by
1989 AIME Problems
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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