Difference between revisions of "1985 AIME Problems"

Latest revision as of 18:24, 23 February 2024

1985 AIME (Answer Key) Printable version \| AoPS Contest Collections • PDF
Instructions 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. 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

Let $x_1=97$ , and for $n>1$ let $x_n=\frac{n}{x_{n-1}}$ . Calculate the product $x_1x_2 \ldots x_8$ .

Solution

Problem 2

When a right triangle is rotated about one leg, the volume of the cone produced is $800\pi \;\textrm{cm}^3$ . When the triangle is rotated about the other leg, the volume of the cone produced is $1920\pi \;\textrm{cm}^3$ . What is the length (in cm) of the hypotenuse of the triangle?

Solution

Problem 3

Find $c$ if $a$ , $b$ , and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$ , where $i^2 = -1$ .

Solution

Problem 4

A small square is constructed inside a square of area $1$ by dividing each side of the unit square into $n$ 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 $n$ if the the area of the small square is exactly $\frac1{1985}$ .

Solution

Problem 5

A sequence of integers $a_1, a_2, a_3, \ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3$ . 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 $1492$ ?

Solution

Problem 6

As shown in the figure, $\triangle ABC$ 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 $\triangle ABC$ .

Solution

Problem 7

Assume that $a$ , $b$ , $c$ and $d$ are positive integers such that $a^5 = b^4$ , $c^3 = d^2$ and $c - a = 19$ . Determine $d - b$ .

Solution

Problem 8

The sum of the following seven numbers is exactly 19: $a_1 = 2.56$ , $a_2 = 2.61$ , $a_3 = 2.65$ , $a_4 = 2.71$ , $a_5 = 2.79$ , $a_6 = 2.82$ , $a_7 = 2.86$ . It is desired to replace each $a_i$ by an integer approximation $A_i$ , $1\le i \le 7$ , so that the sum of the $A_i$ 's is also $19$ and so that $M$ , the maximum of the "errors" $\lvert A_i-a_i \rvert$ , is as small as possible. For this minimum $M$ , what is $100M$ ?

Solution

Problem 9

In a circle, parallel chords of lengths $2$ , $3$ , and $4$ determine central angles of $\alpha$ , $\beta$ , and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$ . If $\cos \alpha$ , which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?

Solution

Problem 10

How many of the first $1000$ positive integers can be expressed in the form

$\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor$ ,

where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$ ?

Solution

Problem 11

An ellipse has foci at $(9, 20)$ and $(49, 55)$ in the $xy$ -plane and is tangent to the $x$ -axis. What is the length of its major axis?

Solution

Problem 12

Let $A$ , $B$ , $C$ and $D$ be the vertices of a regular tetrahedron, each of whose edges measures $1$ meter. A bug, starting from vertex $A$ , 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 $p = \frac{n}{729}$ be the probability that the bug is at vertex $A$ when it has crawled exactly $7$ meters. Find the value of $n$ .

Solution

Problem 13

The numbers in the sequence $101$ , $104$ , $109$ , $116$ , $\ldots$ are of the form $a_n=100+n^2$ , where $n=1,2,3,\ldots$ . For each $n$ , let $d_n$ be the greatest common divisor of $a_n$ and $a_{n+1}$ . Find the maximum value of $d_n$ as $n$ ranges through the positive integers.

Solution

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 $\frac{1}{2}$ 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?

Solution

Problem 15

Three $12$ cm $\times 12$ cm squares are each cut into two pieces $A$ and $B$ , 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 $\mathrm{cm}^3$ ) of this polyhedron?

Solution

@@ Line 1: / Line 1: @@
+{{AIME Problems|year=1985}}
 ==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>x_1x_2x_3x_4x_5x_6x_7x_8</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 \ldots x_8</math>.
 [[1985 AIME Problems/Problem 1 | Solution]]
 ==Problem 2==
 When a right triangle is rotated about one leg, the volume of the cone produced is <math>800\pi \;\textrm{cm}^3</math>. When the triangle is rotated about the other leg, the volume of the cone produced is <math>1920\pi \;\textrm{cm}^3</math>. What is the length (in cm) of the hypotenuse of the triangle?
@@ Line 13: / Line 16: @@
 ==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]]
@@ Line 20: / Line 23: @@
 ==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]]
 ==Problem 6==
-As shown in the figure, triangle <math>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 triangle <math>ABC</math>.
+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]]
@@ Line 32: / Line 35: @@
 ==Problem 7==
-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>.
+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]]
 ==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 = 2.81</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 maximum absolute value of the difference, is as small as possible. For this minimum <math>M</math>, what is <math>100M</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>\lvert A_i-a_i \rvert</math>, is as small as possible. For this minimum <math>M</math>, what is <math>100M</math>?
 [[1985 AIME Problems/Problem 8 | Solution]]
 ==Problem 9==
-In a circle, parallel chords of lengths 2, 3, and 4 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?
+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]]
 ==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>,
@@ Line 61: / Line 64: @@
 ==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 vertices of a regular tetrahedron, each of whose edges 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]]
 ==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]]
 ==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 1/2 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 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?
+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]]
 ==Problem 15==
-Three 12 cm <math>\times</math>12 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?
+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]]
@@ Line 83: / Line 86: @@
 ==See also==
+{{AIME box|year=1985|before=[[1984 AIME Problems]]|after=[[1986 AIME Problems]]}}
 * [[1985 AIME]]
 * [[American Invitational Mathematics Examination]]
@@ Line 88: / Line 94: @@
 * [[Mathematics competition resources]]
+{{MAA Notice}}
 [[Category:AIME Problems]]

1985 AIME (Problems • Answer Key • Resources)
Preceded by 1984 AIME Problems	Followed by 1986 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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Difference between revisions of "1985 AIME Problems"

Latest revision as of 18:24, 23 February 2024

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

See also