Difference between revisions of "2002 AMC 12A Problems"
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Compute the sum of all the roots of | Compute the sum of all the roots of | ||
− | <math> | + | <math>(2x+3)(x-4)+(2x+3)(x-6)=0 </math> |
<math> \mathrm{(A) \ } \frac{7}{2}\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 13 </math> | <math> \mathrm{(A) \ } \frac{7}{2}\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 13 </math> | ||
Line 35: | Line 35: | ||
== Problem 5 == | == Problem 5 == | ||
+ | Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. | ||
+ | |||
+ | <asy> | ||
+ | unitsize(.3cm); | ||
+ | path c=Circle((0,2),1); | ||
+ | filldraw(Circle((0,0),3),grey,black); | ||
+ | filldraw(Circle((0,0),1),white,black); | ||
+ | filldraw(c,white,black); | ||
+ | filldraw(rotate(60)*c,white,black); | ||
+ | filldraw(rotate(120)*c,white,black); | ||
+ | filldraw(rotate(180)*c,white,black); | ||
+ | filldraw(rotate(240)*c,white,black); | ||
+ | filldraw(rotate(300)*c,white,black); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ \pi \qquad \text{(B)}\ 1.5\pi \qquad \text{(C)}\ 2\pi \qquad \text{(D)}\ 3\pi \qquad \text{(E)}\ 3.5\pi</math> | ||
[[2002 AMC 12A Problems/Problem 5|Solution]] | [[2002 AMC 12A Problems/Problem 5|Solution]] | ||
Line 46: | Line 62: | ||
== Problem 7 == | == Problem 7 == | ||
+ | |||
+ | A <math>45^\circ</math> arc of circle A is equal in length to a <math>30^\circ</math> arc of circle B. What is the ratio of circle A's area and circle B's area? | ||
+ | |||
+ | <math>\text{(A)}\ 4/9 \qquad \text{(B)}\ 2/3 \qquad \text{(C)}\ 5/6 \qquad \text{(D)}\ 3/2 \qquad \text{(E)}\ 9/4</math> | ||
[[2002 AMC 12A Problems/Problem 7|Solution]] | [[2002 AMC 12A Problems/Problem 7|Solution]] | ||
== Problem 8 == | == Problem 8 == | ||
+ | Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let <math>B</math> be the total area of the blue triangles, <math>W</math> the total area of the white squares, and <math>R</math> the area of the red square. Which of the following is correct? | ||
+ | |||
+ | <asy> | ||
+ | unitsize(3mm); | ||
+ | fill((-4,-4)--(-4,4)--(4,4)--(4,-4)--cycle,blue); | ||
+ | fill((-2,-2)--(-2,2)--(2,2)--(2,-2)--cycle,red); | ||
+ | path onewhite=(-3,3)--(-2,4)--(-1,3)--(-2,2)--(-3,3)--(-1,3)--(0,4)--(1,3)--(0,2)--(-1,3)--(1,3)--(2,4)--(3,3)--(2,2)--(1,3)--cycle; | ||
+ | path divider=(-2,2)--(-3,3)--cycle; | ||
+ | fill(onewhite,white); | ||
+ | fill(rotate(90)*onewhite,white); | ||
+ | fill(rotate(180)*onewhite,white); | ||
+ | fill(rotate(270)*onewhite,white); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ B = W \qquad \text{(B)}\ W = R \qquad \text{(C)}\ B = R \qquad \text{(D)}\ 3B = 2R \qquad \text{(E)}\ 2R = W</math> | ||
[[2002 AMC 12A Problems/Problem 8|Solution]] | [[2002 AMC 12A Problems/Problem 8|Solution]] | ||
== Problem 9 == | == Problem 9 == | ||
+ | |||
+ | Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files? | ||
+ | |||
+ | <math>\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 14 \qquad \text{(D)}\ 15 \qquad \text{(E)} 16</math> | ||
[[2002 AMC 12A Problems/Problem 9|Solution]] | [[2002 AMC 12A Problems/Problem 9|Solution]] | ||
Line 65: | Line 104: | ||
== Problem 11 == | == Problem 11 == | ||
+ | |||
+ | Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time? | ||
+ | |||
+ | <math>\text{(A)}\ 45 \qquad \text{(B)}\ 48 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 55 \qquad \text{(E)} 58</math> | ||
[[2002 AMC 12A Problems/Problem 11|Solution]] | [[2002 AMC 12A Problems/Problem 11|Solution]] | ||
== Problem 12 == | == Problem 12 == | ||
+ | |||
+ | Both roots of the quadratic equation <math>x^2 - 63x + k = 0</math> are prime numbers. The number of possible values of <math>k</math> is | ||
+ | |||
+ | <math>\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 4 \qquad \text{(E) more than 4}</math> | ||
[[2002 AMC 12A Problems/Problem 12|Solution]] | [[2002 AMC 12A Problems/Problem 12|Solution]] | ||
== Problem 13 == | == Problem 13 == | ||
+ | |||
+ | <math> | ||
+ | \text{(A) } | ||
+ | \qquad | ||
+ | \text{(B) } | ||
+ | \qquad | ||
+ | \text{(C) } | ||
+ | \qquad | ||
+ | \text{(D) } | ||
+ | \qquad | ||
+ | \text{(E) } | ||
+ | </math> | ||
[[2002 AMC 12A Problems/Problem 13|Solution]] | [[2002 AMC 12A Problems/Problem 13|Solution]] | ||
== Problem 14 == | == Problem 14 == | ||
+ | |||
+ | <math> | ||
+ | \text{(A) } | ||
+ | \qquad | ||
+ | \text{(B) } | ||
+ | \qquad | ||
+ | \text{(C) } | ||
+ | \qquad | ||
+ | \text{(D) } | ||
+ | \qquad | ||
+ | \text{(E) } | ||
+ | </math> | ||
[[2002 AMC 12A Problems/Problem 14|Solution]] | [[2002 AMC 12A Problems/Problem 14|Solution]] | ||
== Problem 15 == | == Problem 15 == | ||
+ | |||
+ | <math> | ||
+ | \text{(A) } | ||
+ | \qquad | ||
+ | \text{(B) } | ||
+ | \qquad | ||
+ | \text{(C) } | ||
+ | \qquad | ||
+ | \text{(D) } | ||
+ | \qquad | ||
+ | \text{(E) } | ||
+ | </math> | ||
[[2002 AMC 12A Problems/Problem 15|Solution]] | [[2002 AMC 12A Problems/Problem 15|Solution]] | ||
== Problem 16 == | == Problem 16 == | ||
+ | |||
+ | <math> | ||
+ | \text{(A) } | ||
+ | \qquad | ||
+ | \text{(B) } | ||
+ | \qquad | ||
+ | \text{(C) } | ||
+ | \qquad | ||
+ | \text{(D) } | ||
+ | \qquad | ||
+ | \text{(E) } | ||
+ | </math> | ||
[[2002 AMC 12A Problems/Problem 16|Solution]] | [[2002 AMC 12A Problems/Problem 16|Solution]] | ||
== Problem 17 == | == Problem 17 == | ||
+ | |||
+ | <math> | ||
+ | \text{(A) } | ||
+ | \qquad | ||
+ | \text{(B) } | ||
+ | \qquad | ||
+ | \text{(C) } | ||
+ | \qquad | ||
+ | \text{(D) } | ||
+ | \qquad | ||
+ | \text{(E) } | ||
+ | </math> | ||
[[2002 AMC 12A Problems/Problem 17|Solution]] | [[2002 AMC 12A Problems/Problem 17|Solution]] | ||
== Problem 18 == | == Problem 18 == | ||
+ | |||
+ | <math> | ||
+ | \text{(A) } | ||
+ | \qquad | ||
+ | \text{(B) } | ||
+ | \qquad | ||
+ | \text{(C) } | ||
+ | \qquad | ||
+ | \text{(D) } | ||
+ | \qquad | ||
+ | \text{(E) } | ||
+ | </math> | ||
[[2002 AMC 12A Problems/Problem 18|Solution]] | [[2002 AMC 12A Problems/Problem 18|Solution]] | ||
== Problem 19 == | == Problem 19 == | ||
+ | |||
+ | <math> | ||
+ | \text{(A) } | ||
+ | \qquad | ||
+ | \text{(B) } | ||
+ | \qquad | ||
+ | \text{(C) } | ||
+ | \qquad | ||
+ | \text{(D) } | ||
+ | \qquad | ||
+ | \text{(E) } | ||
+ | </math> | ||
[[2002 AMC 12A Problems/Problem 19|Solution]] | [[2002 AMC 12A Problems/Problem 19|Solution]] | ||
== Problem 20 == | == Problem 20 == | ||
+ | |||
+ | <math> | ||
+ | \text{(A) } | ||
+ | \qquad | ||
+ | \text{(B) } | ||
+ | \qquad | ||
+ | \text{(C) } | ||
+ | \qquad | ||
+ | \text{(D) } | ||
+ | \qquad | ||
+ | \text{(E) } | ||
+ | </math> | ||
[[2002 AMC 12A Problems/Problem 20|Solution]] | [[2002 AMC 12A Problems/Problem 20|Solution]] | ||
== Problem 21 == | == Problem 21 == | ||
+ | |||
+ | <math> | ||
+ | \text{(A) } | ||
+ | \qquad | ||
+ | \text{(B) } | ||
+ | \qquad | ||
+ | \text{(C) } | ||
+ | \qquad | ||
+ | \text{(D) } | ||
+ | \qquad | ||
+ | \text{(E) } | ||
+ | </math> | ||
[[2002 AMC 12A Problems/Problem 21|Solution]] | [[2002 AMC 12A Problems/Problem 21|Solution]] | ||
== Problem 22 == | == Problem 22 == | ||
+ | |||
+ | <math> | ||
+ | \text{(A) } | ||
+ | \qquad | ||
+ | \text{(B) } | ||
+ | \qquad | ||
+ | \text{(C) } | ||
+ | \qquad | ||
+ | \text{(D) } | ||
+ | \qquad | ||
+ | \text{(E) } | ||
+ | </math> | ||
[[2002 AMC 12A Problems/Problem 22|Solution]] | [[2002 AMC 12A Problems/Problem 22|Solution]] | ||
== Problem 23 == | == Problem 23 == | ||
+ | |||
+ | <math> | ||
+ | \text{(A) } | ||
+ | \qquad | ||
+ | \text{(B) } | ||
+ | \qquad | ||
+ | \text{(C) } | ||
+ | \qquad | ||
+ | \text{(D) } | ||
+ | \qquad | ||
+ | \text{(E) } | ||
+ | </math> | ||
[[2002 AMC 12A Problems/Problem 23|Solution]] | [[2002 AMC 12A Problems/Problem 23|Solution]] | ||
== Problem 24 == | == Problem 24 == | ||
+ | |||
+ | <math> | ||
+ | \text{(A) } | ||
+ | \qquad | ||
+ | \text{(B) } | ||
+ | \qquad | ||
+ | \text{(C) } | ||
+ | \qquad | ||
+ | \text{(D) } | ||
+ | \qquad | ||
+ | \text{(E) } | ||
+ | </math> | ||
[[2002 AMC 12A Problems/Problem 24|Solution]] | [[2002 AMC 12A Problems/Problem 24|Solution]] | ||
== Problem 25 == | == Problem 25 == | ||
+ | |||
+ | <math> | ||
+ | \text{(A) } | ||
+ | \qquad | ||
+ | \text{(B) } | ||
+ | \qquad | ||
+ | \text{(C) } | ||
+ | \qquad | ||
+ | \text{(D) } | ||
+ | \qquad | ||
+ | \text{(E) } | ||
+ | </math> | ||
[[2002 AMC 12A Problems/Problem 25|Solution]] | [[2002 AMC 12A Problems/Problem 25|Solution]] | ||
== See also == | == See also == | ||
+ | |||
* [[AMC 12]] | * [[AMC 12]] | ||
* [[AMC 12 Problems and Solutions]] | * [[AMC 12 Problems and Solutions]] | ||
* [[2002 AMC 12A]] | * [[2002 AMC 12A]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] |
Revision as of 15:29, 18 February 2009
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 See also
Problem 1
Compute the sum of all the roots of
Problem 2
Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly?
Problem 3
According to the standard convention for exponentiation,
If the order in which the exponentiations are performed is changed, how many other values are possible?
Problem 4
Find the degree measure of an angle whose complement is 25% of its supplement.
Problem 5
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
Problem 6
For how many positive integers does there exist at least one positive integer n such that
?
infinitely many
Problem 7
A arc of circle A is equal in length to a
arc of circle B. What is the ratio of circle A's area and circle B's area?
Problem 8
Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let be the total area of the blue triangles,
the total area of the white squares, and
the area of the red square. Which of the following is correct?
Problem 9
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?
Problem 10
Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?
Problem 11
Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time?
Problem 12
Both roots of the quadratic equation are prime numbers. The number of possible values of
is
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25