Difference between revisions of "1995 AHSME Problems"
(It will all be in wiki form in a few minutes...) |
(so ugly...) |
||
Line 22: | Line 22: | ||
== Problem 3 == | == Problem 3 == | ||
− | The total in-store price for an appliance is <math>\</math>99.99 | + | The total in-store price for an appliance is <math>\</math>99.99. A television commercial advertises the same product for three easy payments of <math>\</math>29.98 and a one-time shipping and handling charge of <math>\</math>9.98. How much is saved by buying the appliance from the television advertiser? |
A. 6 cents | A. 6 cents | ||
Line 33: | Line 33: | ||
== Problem 4 == | == Problem 4 == | ||
− | If < | + | If <math>M</math> is <math>30 \%</math> of <math>Q</math>, <math>Q</math> is <math>20 \%</math> of <math>P</math>, and <math>N</math> is <math>50 \%</math> of <math>P</math>, then <math>\frac {M}{N} =</math> |
− | A. < | + | A. <math>\frac {3}{250}</math> |
− | B. < | + | B. <math>\frac {3}{25}</math> |
C. 1 | C. 1 | ||
− | D. < | + | D. <math>\frac {6}{5}</math> |
− | E. < | + | E. <math>\frac {4}{3}</math> |
[[1995 AMC 12 Problems/Problem 4|Solution]] | [[1995 AMC 12 Problems/Problem 4|Solution]] | ||
Line 80: | Line 80: | ||
== Problem 8 == | == Problem 8 == | ||
− | In < | + | In <math>\triangle ABC</math>, <math>\angle C = 90^\circ, AC = 6</math> and <math>BC = 8</math>. Points <math>D</math> and <math>E</math> are on <math>\overline{AB}</math> and <math>\overline{BC}</math>, respectively, and <math>\angle BED = 90^\circ</math>. If <math>DE = 4</math>, then <math>BD =</math> |
A. 5 | A. 5 | ||
− | B. < | + | B. <math>\frac {16}{3}</math> |
− | C. < | + | C. <math>\frac {20}{3}</math> |
− | D. < | + | D. <math>\frac {15}{2}</math> |
E. 8 | E. 8 | ||
Line 102: | Line 102: | ||
== Problem 10 == | == Problem 10 == | ||
− | The area of the triangle bounded by the lines < | + | The area of the triangle bounded by the lines <math>y = x, y = - x</math> and <math>y = 6</math> is |
A. 12 | A. 12 | ||
− | B. < | + | B. <math>12 \sqrt 2</math> |
C. 24 | C. 24 | ||
− | D. < | + | D. <math>24 \sqrt 2</math> |
E. 36 | E. 36 | ||
Line 113: | Line 113: | ||
== Problem 11 == | == Problem 11 == | ||
− | How many base 10 four-digit numbers, < | + | How many base 10 four-digit numbers, <math>N = \underline{a} \underline{b} \underline{c} \underline{d}</math>, satisfy all three of the following conditions? |
− | (i) < | + | (i) <math>4,000 \leq N < 6,000;</math> (ii) <math>N</math> is a multiple of 5; (iii) <math>3 \leq b < c \leq 6</math>. |
A. 10 | A. 10 | ||
Line 126: | Line 126: | ||
== Problem 12 == | == Problem 12 == | ||
− | Let < | + | Let <math>f</math> be a linear function with the properties that <math>f(1) \leq f(2), f(3) \geq f(4),</math> and <math>f(5) = 5</math>. Which of the following is true? |
− | A. < | + | A. <math>f(0) < 0</math> |
− | B. < | + | B. <math>f(0) = 0</math> |
− | C. < | + | C. <math>f(1) < f(0) < f( - 1)</math> |
− | D. < | + | D. <math>f(0) = 5</math> |
− | E. < | + | E. <math>f(0) > 5</math> |
[[1995 AMC 12 Problems/Problem 12|Solution]] | [[1995 AMC 12 Problems/Problem 12|Solution]] | ||
== Problem 13 == | == Problem 13 == | ||
− | The addition below is incorrect. The display can be made correct by changing one digit < | + | The addition below is incorrect. The display can be made correct by changing one digit <math>d</math>, wherever it occurs, to another digit <math>e</math>. Find the sum of <math>d</math> and <math>e</math>. |
{{image}} | {{image}} | ||
Line 152: | Line 152: | ||
== Problem 14 == | == Problem 14 == | ||
− | If < | + | If <math>f(x) = ax^4 - bx^2 + x + 5</math> and <math>f( - 3) = 2</math>, then <math>f(3) =</math> |
A. -5 | A. -5 | ||
Line 176: | Line 176: | ||
Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that: | Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that: | ||
− | i. The actual attendance in Atlanta is within < | + | i. The actual attendance in Atlanta is within <math>10 \%</math> of Anita's estimate. |
− | ii. Bob's estimate is within < | + | ii. Bob's estimate is within <math>10 \%</math> of the actual attendance in Boston. |
To the nearest 1,000, the largest possible difference between the numbers attending the two games is | To the nearest 1,000, the largest possible difference between the numbers attending the two games is | ||
Line 190: | Line 190: | ||
== Problem 17 == | == Problem 17 == | ||
− | Given regular pentagon < | + | Given regular pentagon <math>ABCDE</math>, a circle can be drawn that is tangent to <math>\overline{DC}</math> at <math>D</math> and to <math>\overline{AB}</math> at <math>A</math>. The number of degrees in minor arc <math>AD</math> is |
A. 72 | A. 72 | ||
Line 203: | Line 203: | ||
== Problem 18 == | == Problem 18 == | ||
− | Two rays with common endpoint < | + | Two rays with common endpoint <math>O</math> forms a <math>30^\circ</math> angle. Point <math>A</math> lies on one ray, point <math>B</math> on the other ray, and <math>AB = 1</math>. The maximum possible length of <math>OB</math> is |
A. 1 | A. 1 | ||
− | B. < | + | B. <math>\frac {1 + \sqrt {3}}{\sqrt 2}</math> |
− | C. < | + | C. <math>\sqrt 3</math> |
D. 2 | D. 2 | ||
− | E. < | + | E. <math>\frac {4}{\sqrt 3}</math> |
[[1995 AMC 12 Problems/Problem 18|Solution]] | [[1995 AMC 12 Problems/Problem 18|Solution]] | ||
== Problem 19 == | == Problem 19 == | ||
− | Equilateral triangle < | + | Equilateral triangle <math>DEF</math> is inscribed in equilateral triangle <math>ABC</math> such that <math>\overline{DE} \perp \overline{BC}</math>. The reatio of the area of <math>\triangle DEF</math> to the area of <math>\triangle ABC</math> is |
− | A. < | + | A. <math>\frac {1}{6}</math> |
− | B. < | + | B. <math>\frac {1}{4}</math> |
− | C. < | + | C. <math>\frac {1}{3}</math> |
− | D. < | + | D. <math>\frac {2}{5}</math> |
− | E. < | + | E. <math>\frac {1}{2}</math> |
[[1995 AMC 12 Problems/Problem 19|Solution]] | [[1995 AMC 12 Problems/Problem 19|Solution]] | ||
== Problem 20 == | == Problem 20 == | ||
− | If < | + | If <math>a,b</math> and <math>c</math> are three (not necessarily different) numbers chosen randomly and with replacement from the set <math>\{1,2,3,4,5 \}</math>, the probability that <math>ab + c</math> is even is |
− | < | + | <math> \mathrm{(A) \ \frac {2}{5} } \qquad \mathrm{(B) \ \frac {59}{125} } \qquad \mathrm{(C) \ \frac {1}{2} } \qquad \mathrm{(D) \ \frac {64}{125} } \qquad \mathrm{(E) \ \frac {3}{5} } </math> |
[[1995 AMC 12 Problems/Problem 20|Solution]] | [[1995 AMC 12 Problems/Problem 20|Solution]] | ||
Line 251: | Line 251: | ||
− | < | + | <math> \mathrm{(A) \ 459 } \qquad \mathrm{(B) \ 600 } \qquad \mathrm{(C) \ 680 } \qquad \mathrm{(D) \ 720 } \qquad \mathrm{(E) \ 745 } </math> |
[[1995 AMC 12 Problems/Problem 22|Solution]] | [[1995 AMC 12 Problems/Problem 22|Solution]] | ||
== Problem 23 == | == Problem 23 == | ||
− | The sides of a triangle have lengths 11,15, and < | + | The sides of a triangle have lengths 11,15, and <math>k</math>, where <math>k</math> is an integer. For how many values of <math>k</math> is the triangle obtuse? |
− | < | + | <math> \mathrm{(A) \ 5 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 12 } \qquad \mathrm{(D) \ 13 } \qquad \mathrm{(E) \ 14 } </math> |
[[1995 AMC 12 Problems/Problem 23|Solution]] | [[1995 AMC 12 Problems/Problem 23|Solution]] | ||
== Problem 24 == | == Problem 24 == | ||
− | There exist positive integers < | + | There exist positive integers <math>A,B</math> and <math>C</math>, with no common factor greater than 1, such that |
<cmath>A \log_{200} 5 + B \log_{200} 2 = C</cmath> | <cmath>A \log_{200} 5 + B \log_{200} 2 = C</cmath> | ||
− | What is < | + | What is <math>A + B + C</math>? |
− | < | + | <math> \mathrm{(A) \ 6 } \qquad \mathrm{(B) \ 7 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 9 } \qquad \mathrm{(E) \ 10 } </math> |
[[1995 AMC 12 Problems/Problem 24|Solution]] | [[1995 AMC 12 Problems/Problem 24|Solution]] | ||
Line 279: | Line 279: | ||
− | < | + | <math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 12 } </math> |
[[1995 AMC 12 Problems/Problem 25|Solution]] | [[1995 AMC 12 Problems/Problem 25|Solution]] | ||
== Problem 26 == | == Problem 26 == | ||
− | In the figure, < | + | In the figure, <math>\overline{AB}</math> and <math>\overline{CD}</math> are diameters of the circle with center <math>O</math>, <math>\overline{AB} \perp \overline{CD}</math>, and chord <math>\overline{DF}</math> intersects <math>\overline{AB}</math> at <math>E</math>. If <math>DE = 6</math> and <math>EF = 2</math>, then the area of the circle is |
Line 291: | Line 291: | ||
Link: http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62477 | Link: http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62477 | ||
− | < | + | <math> \mathrm{(A) \ 23 \pi } \qquad \mathrm{(B) \ \frac {47}{2} \pi } \qquad \mathrm{(C) \ 24 \pi } \qquad \mathrm{(D) \ \frac {49}{2} \pi } \qquad \mathrm{(E) \ 25 \pi } </math> |
Line 303: | Line 303: | ||
A table really needs to go there... Link: http://www.artofproblemsolving.com/Forum/viewtopic.php?search_id=1526011879&t=62471 | A table really needs to go there... Link: http://www.artofproblemsolving.com/Forum/viewtopic.php?search_id=1526011879&t=62471 | ||
− | Let < | + | Let <math>f(n)</math> denote the sum of the numbers in row <math>n</math>. What is the remainder when <math>f(100)</math> is divided by 100? |
− | < | + | <math> \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 30 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 62 } \qquad \mathrm{(E) \ 74 } </math> |
[[1995 AMC 12 Problems/Problem 27|Solution]] | [[1995 AMC 12 Problems/Problem 27|Solution]] | ||
== Problem 28 == | == Problem 28 == | ||
− | Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length < | + | Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length <math>\sqrt {a}</math> where <math>a</math> is |
− | < | + | <math> \mathrm{(A) \ 144 } \qquad \mathrm{(B) \ 156 } \qquad \mathrm{(C) \ 168 } \qquad \mathrm{(D) \ 176 } \qquad \mathrm{(E) \ 184 } </math> |
[[1995 AMC 12 Problems/Problem 28|Solution]] | [[1995 AMC 12 Problems/Problem 28|Solution]] | ||
== Problem 29 == | == Problem 29 == | ||
− | For how many three-element sets of positive integers < | + | For how many three-element sets of positive integers <math>\{a,b,c\}</math> is it true that <math>a \times b \times c = 2310</math>? |
− | < | + | <math> \mathrm{(A) \ 32 } \qquad \mathrm{(B) \ 36 } \qquad \mathrm{(C) \ 40 } \qquad \mathrm{(D) \ 43 } \qquad \mathrm{(E) \ 45 } </math> |
[[1995 AMC 12 Problems/Problem 29|Solution]] | [[1995 AMC 12 Problems/Problem 29|Solution]] | ||
Line 330: | Line 330: | ||
− | < | + | <math> \mathrm{(A) \ 16 } \qquad \mathrm{(B) \ 17 } \qquad \mathrm{(C) \ 18 } \qquad \mathrm{(D) \ 19 } \qquad \mathrm{(E) \ 20 } </math> |
[[1995 AMC 12 Problems/Problem 30|Solution]] | [[1995 AMC 12 Problems/Problem 30|Solution]] |
Revision as of 10:58, 8 January 2008
Contents
[hide]- 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
Kim earned scores of 87,83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will
A. remain the same B. increase by 1 C. increase by 2 D. increase by 3 E. increase by 4
Problem 2
If , then
A. 1 B. C. 7 D. 49 E. 121
Problem 3
The total in-store price for an appliance is 99.99. A television commercial advertises the same product for three easy payments of 29.98 and a one-time shipping and handling charge of 9.98. How much is saved by buying the appliance from the television advertiser?
A. 6 cents B. 7 cents C. 8 cents D. 9 cents E. 10 cents
Problem 4
If is of , is of , and is of , then
A. B. C. 1 D. E.
Problem 5
A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is
A. 500 thousand B. 5 million C. 50 million D. 500 million E. 5 billion
Problem 6
The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked ?
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
A. A B. B C. C D. D E. E
Problem 7
The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is:
A) 25 B) 8 C) 75 D) 50 E) 100
Problem 8
In , and . Points and are on and , respectively, and . If , then
A. 5 B. C. D. E. 8
Problem 9
Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is
A. 10 B. 12 C. 14 D. 16 E. 18
Problem 10
The area of the triangle bounded by the lines and is
A. 12 B. C. 24 D. E. 36
Problem 11
How many base 10 four-digit numbers, , satisfy all three of the following conditions?
(i) (ii) is a multiple of 5; (iii) .
A. 10 B. 18 C. 24 D. 36 E. 48
Problem 12
Let be a linear function with the properties that and . Which of the following is true?
A. B. C. D. E.
Problem 13
The addition below is incorrect. The display can be made correct by changing one digit , wherever it occurs, to another digit . Find the sum of and .
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
Another table... Link: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=373460#373460
A. 4 B. 6 C. 8 D. 10 E. More than 10
Problem 14
If and , then
A. -5 B. -2 C. 1 D. 3 E. 8
Problem 15
Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point
A. 1 B. 2 C. 3 D. 4 E. 5
Problem 16
Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:
i. The actual attendance in Atlanta is within of Anita's estimate. ii. Bob's estimate is within of the actual attendance in Boston.
To the nearest 1,000, the largest possible difference between the numbers attending the two games is
A. 10,000 B. 11,000 C. 20,000 D. 21,000 E. 22,000
Problem 17
Given regular pentagon , a circle can be drawn that is tangent to at and to at . The number of degrees in minor arc is
A. 72 B. 108 C. 120 D. 135 E. 144
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
Problem 18
Two rays with common endpoint forms a angle. Point lies on one ray, point on the other ray, and . The maximum possible length of is
A. 1 B. C. D. 2 E.
Problem 19
Equilateral triangle is inscribed in equilateral triangle such that . The reatio of the area of to the area of is
A.
B.
C.
D.
E.
Problem 20
If and are three (not necessarily different) numbers chosen randomly and with replacement from the set , the probability that is even is
Problem 21
Two nonadjacent vertices of a rectangle are (4,3) and (-4,-3), and the coordinates of the other two vertices are integers. The number of such rectangles is
A. 1 B. 2 C. 3 D. 4 E. 5
Problem 22
A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths 13,19,20,25 and 31, although this is not necessarily their order around the pentagon. The area of the pentagon is
Problem 23
The sides of a triangle have lengths 11,15, and , where is an integer. For how many values of is the triangle obtuse?
Problem 24
There exist positive integers and , with no common factor greater than 1, such that
What is ?
Problem 25
A list of five positive integers has mean 12 and range 18. The mode and median are both 8. How many different values are possible for the second largest element of the list?
Problem 26
In the figure, and are diameters of the circle with center , , and chord intersects at . If and , then the area of the circle is
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
Link: http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62477
Problem 27
Consider the triangular array of numbers with 0,1,2,3,... along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
A table really needs to go there... Link: http://www.artofproblemsolving.com/Forum/viewtopic.php?search_id=1526011879&t=62471
Let denote the sum of the numbers in row . What is the remainder when is divided by 100?
Problem 28
Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length where is
Problem 29
For how many three-element sets of positive integers is it true that ?
Problem 30
A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is