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]] | + | |
+ | <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 | ||
+ | <center><math>|x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|</math>.</center> | ||
+ | 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 (*). | ||
+ | |||
+ | <center>[[Image:AIME_1988_Problem_06.png]]</center> | ||
[[1988 AIME Problems/Problem 6|Solution]] | [[1988 AIME Problems/Problem 6|Solution]] | ||
== Problem 7 == | == Problem 7 == | ||
+ | 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: | ||
+ | <cmath> | ||
+ | \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*} | ||
+ | </cmath> | ||
+ | Calculate <math>f(14,52)</math>. | ||
[[1988 AIME Problems/Problem 8|Solution]] | [[1988 AIME Problems/Problem 8|Solution]] | ||
== Problem 9 == | == Problem 9 == | ||
+ | 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 == | ||
+ | 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 == | ||
+ | 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> | ||
+ | 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 == | ||
+ | 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>. | ||
+ | |||
+ | <center>[[Image:AIME_1988_Problem_12.png]]</center> | ||
[[1988 AIME Problems/Problem 12|Solution]] | [[1988 AIME Problems/Problem 12|Solution]] | ||
== Problem 13 == | == Problem 13 == | ||
+ | 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 == | ||
+ | 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 | ||
+ | <cmath>12x^2 + bxy + cy^2 + d = 0.</cmath> | ||
+ | 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]] | ||
+ | {{MAA Notice}} |
Latest revision as of 06:35, 7 September 2018
1988 AIME (Answer Key) | AoPS Contest Collections | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
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 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?
Problem 2
For any positive integer , let denote the square of the sum of the digits of . For , let . Find .
Problem 3
Find if .
Problem 4
Suppose that for . Suppose further that
What is the smallest possible value of ?
Problem 5
Let , in lowest terms, be the probability that a randomly chosen positive divisor of is an integer multiple of . Find .
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 (*).
Problem 7
In triangle , , and the altitude from divides into segments of length and . What is the area of triangle ?
Problem 8
The function , defined on the set of ordered pairs of positive integers, satisfies the following properties: Calculate .
Problem 9
Find the smallest positive integer whose cube ends in .
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?
Problem 11
Let be complex numbers. A line in the complex plane is called a mean line for the points if contains points (complex numbers) such that
For the numbers , , , , and , there is a unique mean line with y-intercept . Find the slope of this mean line.
Problem 12
Let be an interior point of triangle and extend lines from the vertices through to the opposite sides. Let , , , and denote the lengths of the segments indicated in the figure. Find the product if and .
Problem 13
Find if and are integers such that is a factor of .
Problem 14
Let be the graph of , and denote by the reflection of in the line . Let the equation of be written in the form Find the product .
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.)
See also
1988 AIME (Problems • Answer Key • Resources) | ||
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 |
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.