Difference between revisions of "1989 AHSME Problems"
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+ | {{AHSME Problems | ||
+ | |year = 1989 | ||
+ | }} | ||
== Problem 1 == | == Problem 1 == | ||
Line 136: | Line 139: | ||
== 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]] | ||
== Problem 20 == | == Problem 20 == | ||
+ | |||
+ | Let <math>x</math> be a real number selected uniformly at random between 100 and 200. If <math>\lfloor {\sqrt{x}} \rfloor = 12</math>, find the probability that <math>\lfloor {\sqrt{100x}} \rfloor = 120</math>. (<math>\lfloor {v} \rfloor</math> means the greatest integer less than or equal to <math>v</math>.) | ||
+ | |||
+ | <math>\text{(A)} \ \frac{2}{25} \qquad \text{(B)} \ \frac{241}{2500} \qquad \text{(C)} \ \frac{1}{10} \qquad \text{(D)} \ \frac{96}{625} \qquad \text{(E)} \ 1</math> | ||
+ | |||
[[1989 AHSME Problems/Problem 20|Solution]] | [[1989 AHSME Problems/Problem 20|Solution]] | ||
== Problem 21 == | == Problem 21 == | ||
+ | |||
+ | A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up 36% of the area of the flag, what percent of the area of the flag is blue? | ||
+ | |||
+ | <asy> | ||
+ | draw((0,0)--(0,5)--(5,5)--(5,0)--(0,0)); | ||
+ | draw((0,1)--(4,5)); | ||
+ | draw((1,0)--(5,4)); | ||
+ | draw((0,4)--(4,0)); | ||
+ | draw((1,5)--(5,1)); | ||
+ | label("blue",(2.5,2.5)); | ||
+ | label("red",(1,1)); | ||
+ | label("red",(1,4)); | ||
+ | label("red",(4,1)); | ||
+ | label("red",(4,4)); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ .5\qquad\text{(B)}\ 1\qquad\text{(C)}\ 2\qquad\text{(D)}\ 3\qquad\text{(E)}\ 6</math> | ||
[[1989 AHSME Problems/Problem 21|Solution]] | [[1989 AHSME Problems/Problem 21|Solution]] | ||
== Problem 22 == | == Problem 22 == | ||
+ | |||
+ | A child has a set of 96 distinct blocks. Each block is one of 2 materials (plastic, wood), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 4 shapes (circle, hexagon, square, triangle). How many blocks in the set are different from the 'plastic medium red circle' in exactly 2 ways? (The 'wood medium red square' is such a block) | ||
+ | |||
+ | (A) 29 (B) 39 (C) 48 (D) 56 (E) 62 | ||
[[1989 AHSME Problems/Problem 22|Solution]] | [[1989 AHSME Problems/Problem 22|Solution]] | ||
== Problem 23 == | == Problem 23 == | ||
+ | |||
+ | A particle moves through the first quadrant as follows. During the first minute it moves from the origin to <math>(1,0)</math>. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive x and y axes, moving one unit of distance parallel to an axis in each minute. At which point will the particle be after exactly 1989 minutes? | ||
+ | |||
+ | <asy> | ||
+ | import graph; | ||
+ | Label f; f.p=fontsize(6); | ||
+ | xaxis(0,3.5,Ticks(f, 1.0)); | ||
+ | yaxis(0,4.5,Ticks(f, 1.0)); | ||
+ | draw((0,0)--(1,0)--(1,1)--(0,1)--(0,2)--(2,2)--(2,0)--(3,0)--(3,3)--(0,3)--(0,4)--(1.5,4),blue+linewidth(2)); | ||
+ | arrow((2,4),dir(180),blue); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ (35,44)\qquad\text{(B)}\ (36,45)\qquad\text{(C)}\ (37,45)\qquad\text{(D)}\ (44,35)\qquad\text{(E)}\ (45,36)</math> | ||
+ | |||
[[1989 AHSME Problems/Problem 23|Solution]] | [[1989 AHSME Problems/Problem 23|Solution]] | ||
== Problem 24 == | == Problem 24 == | ||
+ | |||
+ | Five people are sitting at a round table. Let <math>f\geq 0</math> be the number of people sitting next to at least 1 female and <math>m\geq0</math> be the number of people sitting next to at least one male. The number of possible ordered pairs <math>(f,m)</math> is | ||
+ | |||
+ | <math> \mathrm{(A) \ 7 } \qquad \mathrm{(B) \ 8 } \qquad \mathrm{(C) \ 9 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 11 } </math> | ||
+ | |||
[[1989 AHSME Problems/Problem 24|Solution]] | [[1989 AHSME Problems/Problem 24|Solution]] | ||
== Problem 25 == | == Problem 25 == | ||
+ | |||
+ | In a certain cross country meet between 2 teams of 5 runners each, a runner who finishes in the <math>n</math>th position contributes <math>n</math> to his team's score. The team with the lower score wins. If there are no ties among the runners, how many different winning scores are possible? | ||
+ | |||
+ | (A) 10 (B) 13 (C) 27 (D) 120 (E) 126 | ||
[[1989 AHSME Problems/Problem 25|Solution]] | [[1989 AHSME Problems/Problem 25|Solution]] | ||
== Problem 26 == | == Problem 26 == | ||
+ | |||
+ | A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is | ||
+ | |||
+ | <math> \mathrm{(A) \frac{\sqrt{3}}{12} } \qquad \mathrm{(B) \frac{\sqrt{6}}{16} } \qquad \mathrm{(C) \frac{1}{6} } \qquad \mathrm{(D) \frac{\sqrt{2}}{8} } \qquad \mathrm{(E) \frac{1}{4} } </math> | ||
+ | |||
[[1989 AHSME Problems/Problem 26|Solution]] | [[1989 AHSME Problems/Problem 26|Solution]] | ||
== Problem 27 == | == Problem 27 == | ||
+ | |||
+ | Let <math>n</math> be a positive integer. If the equation <math>2x+2y+z=n</math> has 28 solutions in positive integers <math>x</math>, <math>y</math>, and <math>z</math>, then <math>n</math> must be either | ||
+ | |||
+ | <math>\mathrm{(A)}\ 14 \text{ or } 15\ \mathrm{(B)}\ 15 \text{ or } 16\ \mathrm{(C)}\ 16 \text{ or } 17\ \mathrm{(D)}\ 17 \text{ or } 18\ \mathrm{(E)}\ 18 \text{ or } 19</math> | ||
[[1989 AHSME Problems/Problem 27|Solution]] | [[1989 AHSME Problems/Problem 27|Solution]] | ||
== Problem 28 == | == Problem 28 == | ||
+ | |||
+ | Find the sum of the roots of <math>\tan^2x-9\tan x+1=0</math> that are between <math>x=0</math> and <math>x=2\pi</math> radians. | ||
+ | |||
+ | <math> \mathrm{(A) \frac{\pi}{2} } \qquad \mathrm{(B) \pi } \qquad \mathrm{(C) \frac{3\pi}{2} } \qquad \mathrm{(D) 3\pi } \qquad \mathrm{(E) 4\pi } </math> | ||
[[1989 AHSME Problems/Problem 28|Solution]] | [[1989 AHSME Problems/Problem 28|Solution]] | ||
== Problem 29 == | == Problem 29 == | ||
+ | |||
+ | Find <math>\sum_{k=0}^{49}(-1)^k\binom{99}{2k}</math>, where <math>\binom{n}{j}=\frac{n!}{j!(n-j)!}</math> | ||
+ | |||
+ | (A) <math>-2^{50}</math> (B) <math>-2^{49}</math> (C) 0 (D) <math>2^{49}</math> (E) <math>2^{50}</math> | ||
+ | |||
+ | |||
[[1989 AHSME Problems/Problem 29|Solution]] | [[1989 AHSME Problems/Problem 29|Solution]] | ||
== Problem 30 == | == Problem 30 == | ||
+ | |||
+ | Suppose that 7 boys and 13 girls line up in a row. Let <math>S</math> be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row <math>GBBGGGBGBGGGBGBGGBGG</math> we have that <math>S=12</math>. The average value of <math>S</math> (if all possible orders of these 20 people are considered) is closest to | ||
+ | |||
+ | (A) 9 (B) 10 (C) 11 (D) 12 (E) 13 | ||
[[1989 AHSME Problems/Problem 30|Solution]] | [[1989 AHSME Problems/Problem 30|Solution]] | ||
+ | |||
+ | |||
+ | == See also == | ||
+ | |||
+ | * [[AMC 12 Problems and Solutions]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | |||
+ | {{AHSME box|year=1989|before=[[1988 AHSME]]|after=[[1990 AHSME]]}} | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 12:44, 19 February 2020
1989 AHSME (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 |
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
- 31 See also
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?
Problem 20
Let be a real number selected uniformly at random between 100 and 200. If , find the probability that . ( means the greatest integer less than or equal to .)
Problem 21
A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up 36% of the area of the flag, what percent of the area of the flag is blue?
Problem 22
A child has a set of 96 distinct blocks. Each block is one of 2 materials (plastic, wood), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 4 shapes (circle, hexagon, square, triangle). How many blocks in the set are different from the 'plastic medium red circle' in exactly 2 ways? (The 'wood medium red square' is such a block)
(A) 29 (B) 39 (C) 48 (D) 56 (E) 62
Problem 23
A particle moves through the first quadrant as follows. During the first minute it moves from the origin to . Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive x and y axes, moving one unit of distance parallel to an axis in each minute. At which point will the particle be after exactly 1989 minutes?
Problem 24
Five people are sitting at a round table. Let be the number of people sitting next to at least 1 female and be the number of people sitting next to at least one male. The number of possible ordered pairs is
Problem 25
In a certain cross country meet between 2 teams of 5 runners each, a runner who finishes in the th position contributes to his team's score. The team with the lower score wins. If there are no ties among the runners, how many different winning scores are possible?
(A) 10 (B) 13 (C) 27 (D) 120 (E) 126
Problem 26
A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is
Problem 27
Let be a positive integer. If the equation has 28 solutions in positive integers , , and , then must be either
Problem 28
Find the sum of the roots of that are between and radians.
Problem 29
Find , where
(A) (B) (C) 0 (D) (E)
Problem 30
Suppose that 7 boys and 13 girls line up in a row. Let be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row we have that . The average value of (if all possible orders of these 20 people are considered) is closest to
(A) 9 (B) 10 (C) 11 (D) 12 (E) 13
See also
1989 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1988 AHSME |
Followed by 1990 AHSME | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.