Difference between revisions of "1997 AIME Problems"

Latest revision as of 16:08, 25 June 2020

1997 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

How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?

Solution

Problem 2

The nine horizontal and nine vertical lines on an $8\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Solution

Problem 3

Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number?

Solution

Problem 4

Circles of radii 5, 5, 8, and $m/n$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Solution

Problem 5

The number $r$ can be expressed as a four-place decimal $0.abcd,$ where $a, b, c,$ and $d$ represent digits, any of which could be zero. It is desired to approximate $r$ by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to $r$ is $\frac 27.$ What is the number of possible values for $r$ ?

Solution

Problem 6

Point $B$ is in the exterior of the regular $n$ -sided polygon $A_1A_2\cdots A_n$ , and $A_1A_2B$ is an equilateral triangle. What is the largest value of $n$ for which $A_1$ , $A_n$ , and $B$ are consecutive vertices of a regular polygon?

Solution

Problem 7

A car travels due east at $\frac 23$ miles per minute on a long, straight road. At the same time, a circular storm, whose radius is $51$ miles, moves southeast at $\frac 12\sqrt{2}$ miles per minute. At time $t=0$ , the center of the storm is $110$ miles due north of the car. At time $t=t_1$ minutes, the car enters the storm circle, and at time $t=t_2$ minutes, the car leaves the storm circle. Find $\frac 12(t_1+t_2)$ .

Solution

Problem 8

How many different $4\times 4$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0?

Solution

Problem 9

Given a nonnegative real number $x$ , let $\langle x\rangle$ denote the fractional part of $x$ ; that is, $\langle x\rangle=x-\lfloor x\rfloor$ , where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$ . Suppose that $a$ is positive, $\langle a^{-1}\rangle=\langle a^2\rangle$ , and $2<a^2<3$ . Find the value of $a^{12}-144a^{-1}$ .

Solution

Problem 10

Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-shade combination represented. A set of three cards from the deck is called complementary if all of the following statements are true:

i. Either each of the three cards has a different shape or all three of the card have the same shape.

ii. Either each of the three cards has a different color or all three of the cards have the same color.

iii. Either each of the three cards has a different shade or all three of the cards have the same shade.

How many different complementary three-card sets are there?

Solution

Problem 11

Let $x=\frac{\sum\limits_{n=1}^{44} \cos n^\circ}{\sum\limits_{n=1}^{44} \sin n^\circ}$ . What is the greatest integer that does not exceed $100x$ ?

Solution

Problem 12

The function $f$ defined by $f(x)= \frac{ax+b}{cx+d}$ , where $a$ , $b$ , $c$ and $d$ are nonzero real numbers, has the properties $f(19)=19$ , $f(97)=97$ and $f(f(x))=x$ for all values except $\frac{-d}{c}$ . Find the unique number that is not in the range of $f$ .

Solution

Problem 13

Let $S$ be the set of points in the Cartesian plane that satisfy

$\Big|\big| |x|-2\big|-1\Big|+\Big|\big| |y|-2\big|-1\Big|=1.$

If a model of $S$ were built from wire of negligible thickness, then the total length of wire required would be $a\sqrt{b}$ , where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime number. Find $a+b$ .

Solution

Problem 14

Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0$ . Let $m/n$ be the probability that $\sqrt{2+\sqrt{3}}\le |v+w|$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution

Problem 15

The sides of rectangle $ABCD$ have lengths $10$ and $11$ . An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$ . The maximum possible area of such a triangle can be written in the form $p\sqrt{q}-r$ , where $p$ , $q$ , and $r$ are positive integers, and $q$ is not divisible by the square of any prime number. Find $p+q+r$ .

Solution

@@ Line 1: / Line 1: @@
+{{AIME Problems|year=1997}}
 == Problem 1 ==
 How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?
@@ Line 5: / Line 7: @@
 == Problem 2 ==
-The nine horizontal and nine vertical lines on an <math>8\times8</math> checkeboard form <math>r</math> rectangles, of which <math>s</math> are squares.  The number <math>s/r</math> can be written in the form <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m + n.</math>
+The nine horizontal and nine vertical lines on an <math>8\times8</math> checkerboard form <math>r</math> rectangles, of which <math>s</math> are squares.  The number <math>s/r</math> can be written in the form <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m + n.</math>
 [[1997 AIME Problems/Problem 2|Solution]]
 == Problem 3 ==
-Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit nmber.  This number is exactly nine times the product Sarah should have obtained.  What is the sum of the two-digit number and the three-digit number?
+Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number.  This number is exactly nine times the product Sarah should have obtained.  What is the sum of the two-digit number and the three-digit number?
 [[1997 AIME Problems/Problem 3|Solution]]
@@ Line 20: / Line 22: @@
 == Problem 5 ==
+The number <math>r</math> can be expressed as a four-place decimal <math>0.abcd,</math> where <math>a, b, c,</math> and <math>d</math> represent digits, any of which could be zero.  It is desired to approximate <math>r</math> by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to <math>r</math> is <math>\frac 27.</math>  What is the number of possible values for <math>r</math>?
 [[1997 AIME Problems/Problem 5|Solution]]
 == Problem 6 ==
+Point <math>B</math> is in the exterior of the regular <math>n</math>-sided polygon <math>A_1A_2\cdots A_n</math>, and <math>A_1A_2B</math> is an equilateral triangle. What is the largest value of <math>n</math> for which <math>A_1</math>, <math>A_n</math>, and <math>B</math> are consecutive vertices of a regular polygon?
 [[1997 AIME Problems/Problem 6|Solution]]
 == Problem 7 ==
+A car travels due east at <math>\frac 23</math> miles per minute on a long, straight road. At the same time, a circular storm, whose radius is <math>51</math> miles, moves southeast at <math>\frac 12\sqrt{2}</math> miles per minute. At time <math>t=0</math>, the center of the storm is <math>110</math> miles due north of the car. At time <math>t=t_1</math> minutes, the car enters the storm circle, and at time <math>t=t_2</math> minutes, the car leaves the storm circle. Find <math>\frac 12(t_1+t_2)</math>.
 [[1997 AIME Problems/Problem 7|Solution]]
 == Problem 8 ==
+How many different <math>4\times 4</math> arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0?
 [[1997 AIME Problems/Problem 8|Solution]]
 == Problem 9 ==
+Given a nonnegative real number <math>x</math>, let <math>\langle x\rangle</math> denote the fractional part of <math>x</math>; that is, <math>\langle x\rangle=x-\lfloor x\rfloor</math>, where <math>\lfloor x\rfloor</math> denotes the greatest integer less than or equal to <math>x</math>. Suppose that <math>a</math> is positive, <math>\langle a^{-1}\rangle=\langle a^2\rangle</math>, and <math>2<a^2<3</math>. Find the value of <math>a^{12}-144a^{-1}</math>.
 [[1997 AIME Problems/Problem 9|Solution]]
 == Problem 10 ==
+Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-shade combination represented. A set of three cards from the deck is called complementary if all of the following statements are true:
+i. Either each of the three cards has a different shape or all three of the card have the same shape.
+ii. Either each of the three cards has a different color or all three of the cards have the same color.
+iii. Either each of the three cards has a different shade or all three of the cards have the same shade.
+How many different complementary three-card sets are there?
 [[1997 AIME Problems/Problem 10|Solution]]
 == Problem 11 ==
+Let <math>x=\frac{\sum\limits_{n=1}^{44} \cos n^\circ}{\sum\limits_{n=1}^{44} \sin n^\circ}</math>. What is the greatest integer that does not exceed <math>100x</math>?
 [[1997 AIME Problems/Problem 11|Solution]]
 == Problem 12 ==
+The function <math>f</math> defined by <math>f(x)= \frac{ax+b}{cx+d}</math>, where <math>a</math>, <math>b</math>, <math>c</math> and <math>d</math> are nonzero real numbers, has the properties <math>f(19)=19</math>, <math>f(97)=97</math> and <math>f(f(x))=x</math> for all values except <math>\frac{-d}{c}</math>. Find the unique number that is not in the range of <math>f</math>.
 [[1997 AIME Problems/Problem 12|Solution]]
 == Problem 13 ==
+Let <math>S</math> be the set of points in the Cartesian plane that satisfy <center><math>\Big|\big| |x|-2\big|-1\Big|+\Big|\big| |y|-2\big|-1\Big|=1.</math></center> If a model of <math>S</math> were built from wire of negligible thickness, then the total length of wire required would be <math>a\sqrt{b}</math>, where <math>a</math> and <math>b</math> are positive integers and <math>b</math> is not divisible by the square of any prime number. Find <math>a+b</math>.
 [[1997 AIME Problems/Problem 13|Solution]]
 == Problem 14 ==
+Let <math>v</math> and <math>w</math> be distinct, randomly chosen roots of the equation <math>z^{1997}-1=0</math>. Let <math>m/n</math> be the probability that <math>\sqrt{2+\sqrt{3}}\le |v+w|</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 [[1997 AIME Problems/Problem 14|Solution]]
 == Problem 15 ==
+The sides of rectangle <math>ABCD</math> have lengths <math>10</math> and <math>11</math>. An equilateral triangle is drawn so that no point of the triangle lies outside <math>ABCD</math>. The maximum possible area of such a triangle can be written in the form <math>p\sqrt{q}-r</math>, where <math>p</math>, <math>q</math>, and <math>r</math> are positive integers, and <math>q</math> is not divisible by the square of any prime number. Find <math>p+q+r</math>.
 [[1997 AIME Problems/Problem 15|Solution]]
 == See also ==
+{{AIME box|year=1997|before=[[1996 AIME Problems]]|after=[[1998 AIME Problems]]}}
 * [[American Invitational Mathematics Examination]]
 * [[AIME Problems and Solutions]]
 * [[Mathematics competition resources]]
+{{MAA Notice}}

1997 AIME (Problems • Answer Key • Resources)
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Difference between revisions of "1997 AIME Problems"

Latest revision as of 16:08, 25 June 2020

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