Difference between revisions of "2002 AMC 12P Problems"
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{{AMC12 Problems|year=2002|ab=P}} | {{AMC12 Problems|year=2002|ab=P}} | ||
== Problem 1 == | == Problem 1 == | ||
+ | |||
Which of the following numbers is a perfect square? | Which of the following numbers is a perfect square? | ||
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== Problem 2 == | == Problem 2 == | ||
+ | |||
The function <math>f</math> is given by the table | The function <math>f</math> is given by the table | ||
+ | |||
+ | <cmath> | ||
+ | \begin{tabular}{|c||c|c|c|c|c|} | ||
+ | \hline | ||
+ | x & 1 & 2 & 3 & 4 & 5 \\ | ||
+ | \hline | ||
+ | f(x) & 4 & 1 & 3 & 5 & 2 \\ | ||
+ | \hline | ||
+ | \end{tabular} | ||
+ | </cmath> | ||
If <math>u_0=4</math> and <math>u_{n+1} = f(u_n)</math> for <math>n \ge 0</math>, find <math>u_{2002}</math> | If <math>u_0=4</math> and <math>u_{n+1} = f(u_n)</math> for <math>n \ge 0</math>, find <math>u_{2002}</math> | ||
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== Problem 3 == | == Problem 3 == | ||
+ | |||
The dimensions of a rectangular box in inches are all positive integers and the volume of the box is <math>2002</math> in<math>^3</math>. Find the minimum possible sum of the three dimensions. | The dimensions of a rectangular box in inches are all positive integers and the volume of the box is <math>2002</math> in<math>^3</math>. Find the minimum possible sum of the three dimensions. | ||
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== Problem 4 == | == Problem 4 == | ||
+ | |||
Let <math>a</math> and <math>b</math> be distinct real numbers for which | Let <math>a</math> and <math>b</math> be distinct real numbers for which | ||
<cmath>\frac{a}{b} + \frac{a+10b}{b+10a} = 2.</cmath> | <cmath>\frac{a}{b} + \frac{a+10b}{b+10a} = 2.</cmath> | ||
+ | |||
Find <math>\frac{a}{b}</math> | Find <math>\frac{a}{b}</math> | ||
− | <math>\text{(A)} | + | <math> |
+ | \text{(A) }0.4 | ||
+ | \qquad | ||
+ | \text{(B) }0.5 | ||
+ | \qquad | ||
+ | \text{(C) }0.6 | ||
+ | \qquad | ||
+ | \text{(D) }0.7 | ||
+ | \qquad | ||
+ | \text{(E) }0.8 | ||
+ | </math> | ||
[[2002 AMC 12P Problems/Problem 4|Solution]] | [[2002 AMC 12P Problems/Problem 4|Solution]] | ||
== Problem 5 == | == Problem 5 == | ||
+ | |||
For how many positive integers <math>m</math> is | For how many positive integers <math>m</math> is | ||
<cmath>\frac{2002}{m^2 -2}</cmath> | <cmath>\frac{2002}{m^2 -2}</cmath> | ||
+ | |||
+ | a positive integer? | ||
<math> | <math> | ||
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== Problem 6 == | == Problem 6 == | ||
− | Participation in the local soccer league this year is <math>10</math> | + | |
+ | Participation in the local soccer league this year is <math>10\%</math> higher than last year. The number of males increased by <math>5\%</math> and the number of females increased by <math>20\%</math>. What fraction of the soccer league is now female? | ||
<math> | <math> | ||
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Let <math>f_n (x) = \text{sin}^n x + \text{cos}^n x.</math> For how many <math>x</math> in <math>[0,\pi]</math> is it true that | Let <math>f_n (x) = \text{sin}^n x + \text{cos}^n x.</math> For how many <math>x</math> in <math>[0,\pi]</math> is it true that | ||
+ | |||
+ | <cmath>6f_{4}(x)-4f_{6}(x)=2f_{2}(x)?</cmath> | ||
<math> | <math> | ||
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Let <math>t_n = \frac{n(n+1)}{2}</math> be the <math>n</math>th triangular number. Find | Let <math>t_n = \frac{n(n+1)}{2}</math> be the <math>n</math>th triangular number. Find | ||
− | <cmath>\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + ... + \frac{1}{ | + | <cmath>\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + ... + \frac{1}{t_{2002}}</cmath> |
<math> | <math> | ||
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== Problem 14 == | == Problem 14 == | ||
− | Find <math>i + 2i^2 +3i^3 + ... + 2002i^{2002}.</math> | + | Find <math>i + 2i^2 +3i^3 + . . . + 2002i^{2002}.</math> |
<math> | <math> | ||
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== Problem 15 == | == Problem 15 == | ||
+ | |||
There are <math>1001</math> red marbles and <math>1001</math> black marbles in a box. Let <math>P_s</math> be the probability that two marbles drawn at random from the box are the same color, and let <math>P_d</math> be the probability that they are different colors. Find <math>|P_s-P_d|.</math> | There are <math>1001</math> red marbles and <math>1001</math> black marbles in a box. Let <math>P_s</math> be the probability that two marbles drawn at random from the box are the same color, and let <math>P_d</math> be the probability that they are different colors. Find <math>|P_s-P_d|.</math> | ||
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== Problem 19 == | == Problem 19 == | ||
− | In quadrilateral <math>ABCD</math>, <math>m\angle B = m \angle C = 120^{\circ}, AB=3, BC=4, and CD=5.</math> Find the area of <math>ABCD.</math> | + | In quadrilateral <math>ABCD</math>, <math>m\angle B = m \angle C = 120^{\circ}, AB=3, BC=4,</math> and <math>CD=5.</math> Find the area of <math>ABCD.</math> |
<math> | <math> | ||
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Let <math>f</math> be a real-valued function such that | Let <math>f</math> be a real-valued function such that | ||
− | <cmath>f(x) + 2f(\frac{2002}{x}) =3x</cmath> | + | |
+ | <cmath>f(x) + 2f(\frac{2002}{x}) = 3x</cmath> | ||
+ | |||
+ | for all <math>x>0.</math> Find <math>f(2).</math> | ||
+ | |||
<math> | <math> | ||
\text{(A) }1000 | \text{(A) }1000 | ||
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Let <math>a</math> and <math>b</math> be real numbers greater than <math>1</math> for which there exists a positive real number <math>c,</math> different from <math>1</math>, such that | Let <math>a</math> and <math>b</math> be real numbers greater than <math>1</math> for which there exists a positive real number <math>c,</math> different from <math>1</math>, such that | ||
+ | |||
<cmath>2(\log_a{c} + \log_b{c}) = 9\log_{ab}{c}.</cmath> | <cmath>2(\log_a{c} + \log_b{c}) = 9\log_{ab}{c}.</cmath> | ||
+ | |||
Find the largest possible value of <math>\log_a b.</math> | Find the largest possible value of <math>\log_a b.</math> | ||
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== Problem 22 == | == Problem 22 == | ||
− | Under the new AMC <math>10, 12</math> scoring method, <math>6</math> points are given for each correct answer, <math>2.5</math> points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between <math>0</math> and <math>150</math> can be obtained in only one way, for example, a score of <math>104.5</math> can be obtained with <math>17</math> correct answers, <math>1</math> unanswered question, and <math>7</math> incorrect | + | Under the new AMC <math>10, 12</math> scoring method, <math>6</math> points are given for each correct answer, <math>2.5</math> points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between <math>0</math> and <math>150</math> can be obtained in only one way, for example, the only way to obtain a score of <math>146.5</math> is to have <math>24</math> correct answers and one unanswered question. Some scores can be obtained in exactly two ways, for example, a score of <math>104.5</math> can be obtained with <math>17</math> correct answers, <math>1</math> unanswered question, and <math>7</math> incorrect, and also with <math>12</math> correct answers and <math>13</math> unanswered questions. There are (three) scores that can be obtained in exactly three ways. What is their sum? |
<math> | <math> | ||
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== Problem 25 == | == Problem 25 == | ||
− | Let <math>a</math> and <math>b</math> be real numbers such that <math>\sin{a} + \sin{b} = \sqrt{2}{2} and \cos {a} + \cos {b} = \sqrt{6}{2}.</math> Find <math>\sin{(a+b)}.</math> | + | Let <math>a</math> and <math>b</math> be real numbers such that <math>\sin{a} + \sin{b} = \frac{\sqrt{2}}{2}</math> and <math>\cos {a} + \cos {b} = \frac{\sqrt{6}}{2}.</math> Find <math>\sin{(a+b)}.</math> |
<math> | <math> | ||
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== See also == | == See also == | ||
− | {{AMC12 box|year= | + | {{AMC12 box|year=2002|ab=P|before=[[2001 AMC 12 Problems]]|after=[[2002 AMC 12A Problems]]}} |
* [[AMC 12]] | * [[AMC 12]] | ||
* [[AMC 12 Problems and Solutions]] | * [[AMC 12 Problems and Solutions]] | ||
− | * [[2002 AMC | + | * [[2002 AMC 12P]] |
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 04:03, 14 July 2024
2002 AMC 12P (Answer Key) Printable versions: • 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 |
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
Which of the following numbers is a perfect square?
Problem 2
The function is given by the table
If and
for
, find
Problem 3
The dimensions of a rectangular box in inches are all positive integers and the volume of the box is in
. Find the minimum possible sum of the three dimensions.
Problem 4
Let and
be distinct real numbers for which
Find
Problem 5
For how many positive integers is
a positive integer?
Problem 6
Participation in the local soccer league this year is higher than last year. The number of males increased by
and the number of females increased by
. What fraction of the soccer league is now female?
Problem 7
How many three-digit numbers have at least one and at least one
?
Problem 8
Let be a segment of length
, and let points
and
be located on
such that
and
. Let
and
be points on one of the semicircles with diameter
for which
and
are perpendicular to
. Find
Problem 9
Two walls and the ceiling of a room meet at right angles at point A fly is in the air one meter from one wall, eight meters from the other wall, and nine meters from point
. How many meters is the fly from the ceiling?
Problem 10
Let For how many
in
is it true that
Problem 11
Let be the
th triangular number. Find
Problem 12
For how many positive integers is
a prime number?
Problem 13
What is the maximum value of for which there is a set of distinct positive integers
for which
Problem 14
Find
Problem 15
There are red marbles and
black marbles in a box. Let
be the probability that two marbles drawn at random from the box are the same color, and let
be the probability that they are different colors. Find
Problem 16
The altitudes of a triangle are and
The largest angle in this triangle is
Problem 17
Let An equivalent form of
is
Problem 18
If are real numbers such that
,
and
, find
Problem 19
In quadrilateral ,
and
Find the area of
Problem 20
Let be a real-valued function such that
for all Find
Problem 21
Let and
be real numbers greater than
for which there exists a positive real number
different from
, such that
Find the largest possible value of
Problem 22
Under the new AMC scoring method,
points are given for each correct answer,
points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between
and
can be obtained in only one way, for example, the only way to obtain a score of
is to have
correct answers and one unanswered question. Some scores can be obtained in exactly two ways, for example, a score of
can be obtained with
correct answers,
unanswered question, and
incorrect, and also with
correct answers and
unanswered questions. There are (three) scores that can be obtained in exactly three ways. What is their sum?
Problem 23
The equation has a zero of the form
, where
and
are positive real numbers. Find
Problem 24
Let be a regular tetrahedron and Let
be a point inside the face
Denote by
the sum of the distances from
to the faces
and by
the sum of the distances from
to the edges
Then
equals
Problem 25
Let and
be real numbers such that
and
Find
See also
2002 AMC 12P (Problems • Answer Key • Resources) | |
Preceded by 2001 AMC 12 Problems |
Followed by 2002 AMC 12A Problems |
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 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.