# Difference between revisions of "1989 AHSME Problems"

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(Problems 9-19) |
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== Problem 9 == | == Problem 9 == | ||

+ | |||

+ | Mr. and Mrs. Zeta want to name their baby Zeta so that its monogram (first, middle, and last initials) will be in alphabetical order with no letter repeated. How many such monograms are possible? | ||

+ | |||

+ | <math> \textrm{(A)}\ 276\qquad\textrm{(B)}\ 300\qquad\textrm{(C)}\ 552\qquad\textrm{(D)}\ 600\qquad\textrm{(E)}\ 15600 </math> | ||

[[1989 AHSME Problems/Problem 9|Solution]] | [[1989 AHSME Problems/Problem 9|Solution]] | ||

== Problem 10 == | == Problem 10 == | ||

+ | |||

+ | Consider the sequence defined recursively by <math> u_{1}= a </math> (any positive integer), and <math> u_{n+1}=\frac{-1}{u_{n}+1}</math>, <math>n = 1,2,3,\cdots </math>. For which of the following values of <math>n</math> must <math>u_{n}=a</math>? | ||

+ | |||

+ | <math> \textrm{(A)}\ 14\qquad\textrm{(B)}\ 15\qquad\textrm{(C)}\ 16\qquad\textrm{(D)}\ 17\qquad\textrm{(E)}\ 18 </math> | ||

[[1989 AHSME Problems/Problem 10|Solution]] | [[1989 AHSME Problems/Problem 10|Solution]] | ||

== Problem 11 == | == Problem 11 == | ||

+ | |||

+ | Let <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> be positive integers with <math> a < 2b</math>, <math>b < 3c </math>, and <math>c<4d</math>. If <math>d<100</math>, the largest possible value for <math>a</math> is | ||

+ | |||

+ | <math> \textrm{(A)}\ 2367\qquad\textrm{(B)}\ 2375\qquad\textrm{(C)}\ 2391\qquad\textrm{(D)}\ 2399\qquad\textrm{(E)}\ 2400 </math> | ||

[[1989 AHSME Problems/Problem 11|Solution]] | [[1989 AHSME Problems/Problem 11|Solution]] | ||

== Problem 12 == | == Problem 12 == | ||

+ | |||

+ | The traffic on a certain east-west highway moves at a constant speed of 60 miles per hour in both directions. An eastbound driver passes 20 west-bound vehicles in a five-minute interval. Assume vehicles in the westbound lane are equally spaced. Which of the following is closest to the number of westbound vehicles present in a 100-mile section of highway? | ||

+ | |||

+ | <math> \textrm{(A)}\ 100\qquad\textrm{(B)}\ 120\qquad\textrm{(C)}\ 200\qquad\textrm{(D)}\ 240\qquad\textrm{(E)}\ 400 </math> | ||

[[1989 AHSME Problems/Problem 12|Solution]] | [[1989 AHSME Problems/Problem 12|Solution]] | ||

== Problem 13 == | == Problem 13 == | ||

+ | |||

+ | Two strips of width 1 overlap at an angle of <math>\alpha</math> as shown. The area of the overlap (shown shaded) is | ||

+ | |||

+ | <asy> | ||

+ | pair a = (0,0),b= (6,0),c=(0,1),d=(6,1); | ||

+ | transform t = rotate(-45,(3,.5)); | ||

+ | pair e = t*a,f=t*b,g=t*c,h=t*d; | ||

+ | pair i = intersectionpoint(a--b,e--f),j=intersectionpoint(a--b,g--h),k=intersectionpoint(c--d,e--f),l=intersectionpoint(c--d,g--h); | ||

+ | draw(a--b^^c--d^^e--f^^g--h); | ||

+ | filldraw(i--j--l--k--cycle,blue); | ||

+ | label("$\alpha$",i+(-.5,.2)); | ||

+ | //commented out labeling because it doesn't look right. | ||

+ | //path lbl1 = (a+(.5,.2))--(c+(.5,-.2)); | ||

+ | //draw(lbl1); | ||

+ | //label("$1$",lbl1);</asy> | ||

+ | |||

+ | <math> \textrm{(A)}\ \sin\alpha\qquad\textrm{(B)}\ \frac{1}{\sin\alpha}\qquad\textrm{(C)}\ \frac{1}{1-\cos\alpha}\qquad\textrm{(D)}\ \frac{1}{\sin^{2}\alpha}\qquad\textrm{(E)}\ \frac{1}{(1-\cos\alpha)^{2}} </math> | ||

[[1989 AHSME Problems/Problem 13|Solution]] | [[1989 AHSME Problems/Problem 13|Solution]] | ||

== Problem 14 == | == Problem 14 == | ||

+ | |||

+ | <math> \cot 10+\tan 5 = </math> | ||

+ | |||

+ | <math> \textrm{(A)}\ \csc 5\qquad\textrm{(B)}\ \csc 10\qquad\textrm{(C)}\ \sec 5\qquad\textrm{(D)}\ \sec 10\qquad\textrm{(E)}\ \sin 15 </math> | ||

[[1989 AHSME Problems/Problem 14|Solution]] | [[1989 AHSME Problems/Problem 14|Solution]] | ||

== Problem 15 == | == Problem 15 == | ||

+ | |||

+ | In <math>\triangle ABC</math>, <math>AB=5</math>, <math>BC=7</math>, <math>AC=9</math>, and <math>D</math> is on <math>\overline{AC}</math> with <math>BD=5</math>. Find the ratio of <math>AD:DC</math>. | ||

+ | |||

+ | <asy> | ||

+ | draw((3,4)--(0,0)--(9,0)--(3,4)--(6,0)); | ||

+ | dot((0,0)); | ||

+ | dot((9,0)); | ||

+ | dot((3,4)); | ||

+ | dot((6,0)); | ||

+ | label("A", (0,0), W); | ||

+ | label("B", (3,4), N); | ||

+ | label("C", (9,0), E); | ||

+ | label("D", (6,0), S);</asy> | ||

+ | |||

+ | <math> \textrm{(A)}\ 4:3\qquad\textrm{(B)}\ 7:5\qquad\textrm{(C)}\ 11:6\qquad\textrm{(D)}\ 13:5\qquad\textrm{(E)}\ 19:8 </math> | ||

[[1989 AHSME Problems/Problem 15|Solution]] | [[1989 AHSME Problems/Problem 15|Solution]] | ||

== Problem 16 == | == Problem 16 == | ||

+ | |||

+ | A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are (3,17) and (48,281)? (Include both endpoints of the segment in your count.) | ||

+ | |||

+ | <math> \textrm{(A)}\ 2\qquad\textrm{(B)}\ 4\qquad\textrm{(C)}\ 6\qquad\textrm{(D)}\ 16\qquad\textrm{(E)}\ 46 </math> | ||

[[1989 AHSME Problems/Problem 16|Solution]] | [[1989 AHSME Problems/Problem 16|Solution]] | ||

== Problem 17 == | == Problem 17 == | ||

+ | |||

+ | The perimeter of an equilateral triangle exceeds the perimeter of a square by <math>1989</math> cm. The length of each side of the triangle exceeds the length of each side of the square by <math>d</math> cm. The square has perimeter greater than 0. How many positive integers are NOT a possible value for <math>d</math>? | ||

+ | |||

+ | <math> \textrm{(A)}\ 0\qquad\textrm{(B)}\ 9\qquad\textrm{(C)}\ 221\qquad\textrm{(D)}\ 663\qquad\textrm{(E)}\ \text{infinitely many} </math> | ||

[[1989 AHSME Problems/Problem 17|Solution]] | [[1989 AHSME Problems/Problem 17|Solution]] | ||

== Problem 18 == | == Problem 18 == | ||

+ | |||

+ | The set of all numbers x for which <math> x+\sqrt{x^{2}+1}-\frac{1}{x+\sqrt{x^{2}+1}} </math> is a rational number is the set of all: | ||

+ | |||

+ | <math> \textrm{(A)}\ \text{ integers }x\qquad\textrm{(B)}\ \text{ rational }x\qquad\textrm{(C)}\ \text{ real }x\qquad\textrm{(D)}\ x\text{ for which }\sqrt{x^{2}+1}\text{ is rational}\qquad\textrm{(E)}\ x\text{ for which }x+\sqrt{x^{2}+1}\text{ is rational } </math> | ||

[[1989 AHSME Problems/Problem 18|Solution]] | [[1989 AHSME Problems/Problem 18|Solution]] | ||

== Problem 19 == | == Problem 19 == | ||

+ | |||

+ | A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths <math>3</math>, <math>4</math>, and <math>5</math>. What is the area of the triangle? | ||

+ | |||

+ | <math> \textrm{(A)}\ 6\qquad\textrm{(B)}\ \frac{18}{\pi^{2}}\qquad\textrm{(C)}\ \frac{9}{\pi^{2}}\left(\sqrt{3}-1\right)\qquad\textrm{(D)}\ \frac{9}{\pi^{2}}\left(\sqrt{3}+1\right)\qquad\textrm{(E)}\ \frac{9}{\pi^{2}}\left(\sqrt{3}+3\right) </math> | ||

[[1989 AHSME Problems/Problem 19|Solution]] | [[1989 AHSME Problems/Problem 19|Solution]] |

## Revision as of 17:14, 5 February 2012

## Contents

- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30

## Problem 1

## Problem 2

## Problem 3

A square is cut into three rectangles along two lines parallel to a side, as shown. If the perimeter of each of the three rectangles is 24, then the area of the original square is

## Problem 4

In the figure, is an isosceles trapezoid with side lengths , , and . The point is on and is the midpoint of hypotenuse in right triangle . Then

## Problem 5

Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpicks wide, then the number of toothpicks used is

## Problem 6

If and the triangle in the first quadrant bounded by the coordinate axes and the graph of has area , then

## Problem 7

In , , , , is an altitude, and is a median. Then

## Problem 8

For how many integers between and does factor into the product of two linear factors with integer coefficients?

## Problem 9

Mr. and Mrs. Zeta want to name their baby Zeta so that its monogram (first, middle, and last initials) will be in alphabetical order with no letter repeated. How many such monograms are possible?

## Problem 10

Consider the sequence defined recursively by (any positive integer), and , . For which of the following values of must ?

## Problem 11

Let , , , and be positive integers with , , and . If , the largest possible value for is

## Problem 12

The traffic on a certain east-west highway moves at a constant speed of 60 miles per hour in both directions. An eastbound driver passes 20 west-bound vehicles in a five-minute interval. Assume vehicles in the westbound lane are equally spaced. Which of the following is closest to the number of westbound vehicles present in a 100-mile section of highway?

## Problem 13

Two strips of width 1 overlap at an angle of as shown. The area of the overlap (shown shaded) is

## Problem 14

## Problem 15

In , , , , and is on with . Find the ratio of .

## Problem 16

A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are (3,17) and (48,281)? (Include both endpoints of the segment in your count.)

## Problem 17

The perimeter of an equilateral triangle exceeds the perimeter of a square by cm. The length of each side of the triangle exceeds the length of each side of the square by cm. The square has perimeter greater than 0. How many positive integers are NOT a possible value for ?

## Problem 18

The set of all numbers x for which is a rational number is the set of all:

## Problem 19

A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths , , and . What is the area of the triangle?