Difference between revisions of "1985 AIME Problems"
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==Problem 1== | ==Problem 1== | ||
− | Let <math>x_1=97</math>, and for <math>n>1</math> let <math>x_n=\frac{n}{x_{n-1}}</math>. Calculate the product <math> | + | Let <math>x_1=97</math>, and for <math>n>1</math> let <math>x_n=\frac{n}{x_{n-1}}</math>. Calculate the product <math>x_1x_2...x_8</math>. |
[[1985 AIME Problems/Problem 1 | Solution]] | [[1985 AIME Problems/Problem 1 | Solution]] | ||
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==Problem 4== | ==Problem 4== | ||
− | A small square is constructed inside a square of area 1 by dividing each side of the unit square into <math>n</math> equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of <math>n</math> if the the area of the small square is exactly <math>\frac1{1985}</math>. | + | A small square is constructed inside a square of area <math>1</math> by dividing each side of the unit square into <math>n</math> equal parts, and then connecting the vertices to the division points closest to the opposite vertices, as shown in the figure. Find the value of <math>n</math> if the the area of the small square is exactly <math>\frac1{1985}</math>. |
[[Image:AIME 1985 Problem 4.png]] | [[Image:AIME 1985 Problem 4.png]] | ||
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==Problem 5== | ==Problem 5== | ||
− | A sequence of integers <math>a_1, a_2, a_3, \ldots</math> is chosen so that <math>a_n = a_{n - 1} - a_{n - 2}</math> for each <math>n \ge 3</math>. What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492? | + | A sequence of integers <math>a_1, a_2, a_3, \ldots</math> is chosen so that <math>a_n = a_{n - 1} - a_{n - 2}</math> for each <math>n \ge 3</math>. What is the sum of the first <math>2001</math> terms of this sequence if the sum of the first <math>1492</math> terms is <math>1985</math>, and the sum of the first <math>1985</math> terms is <math>1492</math>? |
[[1985 AIME Problems/Problem 5 | Solution]] | [[1985 AIME Problems/Problem 5 | Solution]] | ||
==Problem 6== | ==Problem 6== | ||
− | As shown in the figure, | + | As shown in the figure, <math>\triangle ABC</math> is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of <math>\triangle ABC</math>. |
[[Image:AIME 1985 Problem 6.png]] | [[Image:AIME 1985 Problem 6.png]] | ||
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==Problem 7== | ==Problem 7== | ||
− | Assume that <math>a</math>, <math>b</math>, <math>c</math> | + | Assume that <math>a</math>, <math>b</math>, <math>c</math> and <math>d</math> are positive integers such that <math>a^5 = b^4</math>, <math>c^3 = d^2</math> and <math>c - a = 19</math>. Determine <math>d - b</math>. |
[[1985 AIME Problems/Problem 7 | Solution]] | [[1985 AIME Problems/Problem 7 | Solution]] | ||
==Problem 8== | ==Problem 8== | ||
− | The sum of the following seven numbers is exactly 19: <math>a_1 = 2.56</math>, <math>a_2 = 2.61</math>, <math>a_3 = 2.65</math>, <math>a_4 = 2.71</math>, <math>a_5 = 2.79</math>, <math>a_6 = 2.82</math>, <math>a_7 = 2.86</math>. It is desired to replace each <math>a_i</math> by an integer approximation <math>A_i</math>, <math>1\le i \le 7</math>, so that the sum of the <math>A_i</math>'s is also 19 and so that <math>M</math>, the maximum of the "errors" <math>\| A_i-a_i\|</math> | + | The sum of the following seven numbers is exactly 19: <math>a_1 = 2.56</math>, <math>a_2 = 2.61</math>, <math>a_3 = 2.65</math>, <math>a_4 = 2.71</math>, <math>a_5 = 2.79</math>, <math>a_6 = 2.82</math>, <math>a_7 = 2.86</math>. It is desired to replace each <math>a_i</math> by an integer approximation <math>A_i</math>, <math>1\le i \le 7</math>, so that the sum of the <math>A_i</math>'s is also <math>19</math> and so that <math>M</math>, the maximum of the "errors" <math>\| A_i-a_i\|</math>, is as small as possible. For this minimum <math>M</math>, what is <math>100M</math>? |
[[1985 AIME Problems/Problem 8 | Solution]] | [[1985 AIME Problems/Problem 8 | Solution]] | ||
==Problem 9== | ==Problem 9== | ||
− | In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of <math>\alpha</math>, <math>\beta</math> | + | In a circle, parallel chords of lengths <math>2</math>, <math>3</math>, and <math>4</math> determine central angles of <math>\alpha</math>, <math>\beta</math> and <math>\alpha + \beta </math> radians, respectively, where <math>\alpha + \beta < \pi</math>. If <math>\cos \alpha</math>, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator? |
[[1985 AIME Problems/Problem 9 | Solution]] | [[1985 AIME Problems/Problem 9 | Solution]] | ||
==Problem 10== | ==Problem 10== | ||
− | How many of the first 1000 positive integers can be expressed in the form | + | How many of the first <math>1000</math> positive integers can be expressed in the form |
<math>\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor</math>, | <math>\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor</math>, | ||
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==Problem 12== | ==Problem 12== | ||
− | Let <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math> be the [[vertex | vertices]] of a regular [[tetrahedron]] each of whose [[edge]]s measures 1 meter. A bug, starting from vertex <math>A</math>, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let <math>p = \frac n{729}</math> be the [[probability]] that the bug is at vertex <math>A</math> when it has crawled exactly 7 meters. Find the value of <math>n</math>. | + | Let <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math> be the [[vertex | vertices]] of a regular [[tetrahedron]], each of whose [[edge]]s measures <math>1</math> meter. A bug, starting from vertex <math>A</math>, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let <math>p = \frac{n}{729}</math> be the [[probability]] that the bug is at vertex <math>A</math> when it has crawled exactly <math>7</math> meters. Find the value of <math>n</math>. |
[[1985 AIME Problems/Problem 12 | Solution]] | [[1985 AIME Problems/Problem 12 | Solution]] | ||
==Problem 13== | ==Problem 13== | ||
− | The numbers in the [[sequence]] <math>101</math>, <math>104</math>, <math>109</math>, <math>116</math>,<math>\ldots</math> are of the form <math>a_n=100+n^2</math>, where <math>n=1,2,3,\ldots</math> For each <math>n</math>, let <math>d_n</math> be the greatest common divisor of <math>a_n</math> and <math>a_{n+1}</math>. Find the maximum value of <math>d_n</math> as <math>n</math> ranges through the [[positive integer]]s. | + | The numbers in the [[sequence]] <math>101</math>, <math>104</math>, <math>109</math>, <math>116</math>,<math>\ldots</math> are of the form <math>a_n=100+n^2</math>, where <math>n=1,2,3,\ldots</math>. For each <math>n</math>, let <math>d_n</math> be the greatest common divisor of <math>a_n</math> and <math>a_{n+1}</math>. Find the maximum value of <math>d_n</math> as <math>n</math> ranges through the [[positive integer]]s. |
[[1985 AIME Problems/Problem 13 | Solution]] | [[1985 AIME Problems/Problem 13 | Solution]] | ||
==Problem 14== | ==Problem 14== | ||
− | In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 1/ | + | In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned <math>\frac{1}{2}</math> point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned in games against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament? |
[[1985 AIME Problems/Problem 14 | Solution]] | [[1985 AIME Problems/Problem 14 | Solution]] | ||
==Problem 15== | ==Problem 15== | ||
− | Three 12 cm <math>\times</math> | + | Three <math>12</math> cm <math>\times 12</math> cm [[square (geometry) | squares]] are each cut into two pieces <math>A</math> and <math>B</math>, as shown in the first figure below, by joining the [[midpoint]]s of two adjacent sides. These six pieces are then attached to a [[regular polygon | regular]] [[hexagon]], as shown in the second figure, so as to fold into a [[polyhedron]]. What is the [[volume]] (in <math>\mathrm{cm}^3</math>) of this polyhedron? |
[[Image:AIME 1985 Problem 15.png]] | [[Image:AIME 1985 Problem 15.png]] |
Revision as of 12:46, 3 April 2018
1985 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
Let , and for let . Calculate the product .
Problem 2
When a right triangle is rotated about one leg, the volume of the cone produced is . When the triangle is rotated about the other leg, the volume of the cone produced is . What is the length (in cm) of the hypotenuse of the triangle?
Problem 3
Find if , , and are positive integers which satisfy , where .
Problem 4
A small square is constructed inside a square of area by dividing each side of the unit square into equal parts, and then connecting the vertices to the division points closest to the opposite vertices, as shown in the figure. Find the value of if the the area of the small square is exactly .
Problem 5
A sequence of integers is chosen so that for each . What is the sum of the first terms of this sequence if the sum of the first terms is , and the sum of the first terms is ?
Problem 6
As shown in the figure, is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of .
Problem 7
Assume that , , and are positive integers such that , and . Determine .
Problem 8
The sum of the following seven numbers is exactly 19: , , , , , , . It is desired to replace each by an integer approximation , , so that the sum of the 's is also and so that , the maximum of the "errors" , is as small as possible. For this minimum , what is ?
Problem 9
In a circle, parallel chords of lengths , , and determine central angles of , and radians, respectively, where . If , which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?
Problem 10
How many of the first positive integers can be expressed in the form
,
where is a real number, and denotes the greatest integer less than or equal to ?
Problem 11
An ellipse has foci at and in the -plane and is tangent to the -axis. What is the length of its major axis?
Problem 12
Let , , and be the vertices of a regular tetrahedron, each of whose edges measures meter. A bug, starting from vertex , observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let be the probability that the bug is at vertex when it has crawled exactly meters. Find the value of .
Problem 13
The numbers in the sequence , , , , are of the form , where . For each , let be the greatest common divisor of and . Find the maximum value of as ranges through the positive integers.
Problem 14
In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned in games against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?
Problem 15
Three cm cm squares are each cut into two pieces and , as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in ) of this polyhedron?
See also
- 1985 AIME
- 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.