Difference between revisions of "2002 AMC 12P Problems"

Revision as of 21:50, 29 December 2023

2002 AMC 12P (Answer Key) Printable versions: Wiki • AoPS Resources • PDF
Instructions This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator). Figures are not necessarily drawn to scale. You will have 75 minutes working time to complete the test.
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

Problem 1

Which of the following numbers is a perfect square?

$\text{(A)}\ 4^5 5^5 6^6 \qquad \text{(B)}\ 4^4 5^6 6^5 \qquad \text{(C)}\ 4^5 5^4 6^6 \qquad\text{(D)} 4^6 5^4 6^5 \qquad \text{(E)}\ 4^6 5^5 6^4$

Solution

Problem 2

The function $f$ is given by the table

If $u_0=4$ and $u_{n+1} = f(u_n)$ for $n>=0$ , find $u_2002$

$\text{(A)}\ 1\qquad \text{(B)}\ 2\qquad \text{(C)}\ 3\qquad \text{(D)}\ 4\qquad \text{(E)}\ 5$

Solution

Problem 3

The dimensions of a rectangular box in inches are all positive integers and the volume of the box is $2002$ in^3. Find the minimum possible sum of the three dimensions.

$\text{(A) }36 \qquad \text{(B) }38 \qquad \text{(C) }42 \qquad \text{(D) }44 \qquad \text{(E) }92$

Solution

Problem 4

Let $a$ and $b$ be distinct real numbers for which $\frac{a}{b} + \frac{a+10b}{b+10a} = 2.$ Find $\frac{a}{b}$

$\text{(A)}\ 0.4\qquad \text{(B)}\ 0.5\qquad \text{(C)}\ 0.6\qquad \text{(D)}\ 0.7\qquad \text{(E)}\ 0.8$

Solution

Problem 5

For how many positive integers $m$ is $\frac{2002}{m^2 -2}$

$\text{(A)}\ one\qquad \text{(B)}\ two\qquad \text{(C)}\ three\qquad \text{(D)}\ four\qquad \text{(E)}\ five$

Solution

Problem 6

Participation in the local soccer league this year is $10%$ (Error compiling LaTeX. Unknown error_msg) higher than last year. The number of males increased by $5%$ (Error compiling LaTeX. Unknown error_msg) and the number of females increased by $20%$ (Error compiling LaTeX. Unknown error_msg). What fraction of the soccer league is now female?

$\text{(A) }\frac{1}{3} \qquad \text{(B) }\frac{4}{11} \qquad \text{(C) }\frac{2}{5} \qquad \text{(D) }\frac{4}{9} \qquad \text{(E) }\frac{1}{2}$

Solution

Problem 7

How many three-digit numbers have at least one 2 and at least one 3?

$\text{(A) }52 \qquad \text{(B) }54 \qquad \text{(C) }56 \qquad \text{(D) }58 \qquad \text{(E) }60$

Solution

Problem 8

Let $AB$ be a segment of length $26$ , and let points $C$ and $D$ be located on $AB$ such that $AC=1$ and $AD=8$ . Let $E$ and $F$ be points on one of the semicircles with diameter $AB$ for which $EC$ and $FD$ are perpendicular to $AB$ . Find $EF.$

$\text{(A) }5 \qquad \text{(B) }5 \sqrt{2} \qquad \text{(C) }7 \qquad \text{(D) }7 \sqrt{2} \qquad \text{(E) }12$ Solution

Problem 9

Two walls and the ceiling of a room meet at right angles at point $P.$ A fly is in the air one meter from one wall, eight meters from the other wall, and nine meters from point $P$ . How many meters is the fly from the ceiling?

$\text{(A)}\ \sqrt{13} \qquad \text{(B)}\ 2 \qquad \text{(C)}\ \frac52 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ \frac{18}5$

Solution

Problem 10

Let $f_n (x) = sin^n x + cos^n x.$ For how many $x$ in [$0,π$ (Error compiling LaTeX. Unknown error_msg)] $is it true that$ \text{(A) }2 \qquad \text{(B) }4 \qquad \text{(C) }6 \qquad \text{(D) }8 \qquad \text{(E) }more than 8 $[[2002 AMC 12P Problems/Problem 10|Solution]]

== Problem 11 ==

Let$ (Error compiling LaTeX. Unknown error_msg)t_n = \frac{n(n+1)}{2} $be the$ n$th triangular number. Find

<cmath>\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + ... + \frac{1}{t_2002}</cmath>$ (Error compiling LaTeX. Unknown error_msg) \text{(A) }\frac {4003}{2003} \qquad \text{(B) }\frac {2001}{1001} \qquad \text{(C) }\frac {4004}{2003} \qquad \text{(D) }\frac {4001}{2001} \qquad \text{(E) }2 $[[2002 AMC 12P Problems/Problem 11|Solution]]

== Problem 12 ==

For how many positive integers$ (Error compiling LaTeX. Unknown error_msg)n $is$ n^3 - 8n^2 + 20n - 13 $a prime number?$ \text{(A) }one \qquad \text{(B) }two \qquad \text{(C) }three \qquad \text{(D) }four \qquad \text{(E) }more than four $[[2002 AMC 12P Problems/Problem 12|Solution]]

== Problem 13 ==

What is the maximum value of$ (Error compiling LaTeX. Unknown error_msg)n $for which there is a set of distinct positive integers$ k_1, k_2, ... k_n $for which$ k^2_1 + k^2_2 + ... + k^2_n = 2002.$$ (Error compiling LaTeX. Unknown error_msg) \text{(A) }14 \qquad \text{(B) }15 \qquad \text{(C) }16 \qquad \text{(D) }17 \qquad \text{(E) }18 $[[2002 AMC 12P Problems/Problem 13|Solution]]

== Problem 14 ==

Find$ (Error compiling LaTeX. Unknown error_msg)i + 2i^2 +3i^3 + ... + 2002i^2002.$$ (Error compiling LaTeX. Unknown error_msg) \text{(A) }-999 + 1002i \qquad \text{(B) }-1002 + 999i \qquad \text{(C) }-1001 + 1000i \qquad \text{(D) }-1002 + 1001i \qquad \text{(E) }i $[[2002 AMC 12P Problems/Problem 14|Solution]]

== Problem 15 == There are$ (Error compiling LaTeX. Unknown error_msg)1001 red marbles and $1001 black marbles in a box. Let$ P_s $be the probability that two marbles drawn at random from the box are the same color, and let$ P_d $be the probability that they are different colors. Find$ |P_s-P_d|.$$ (Error compiling LaTeX. Unknown error_msg) \text{(A) }0 \qquad \text{(B) }\frac{1}{2002} \qquad \text{(C) }\frac{1}{2001} \qquad \text{(D) }\frac {2}{2001} \qquad \text{(E) }\frac{1}{1000} $[[2002 AMC 12P Problems/Problem 15|Solution]]

== Problem 16 ==

The altitudes of a triangle are$ (Error compiling LaTeX. Unknown error_msg)12, 15, $and$ 20. $The largest angle in this triangle is$ \text{(A) }72^o \qquad \text{(B) }75^o \qquad \text{(C) }90^o \qquad \text{(D) }108^o \qquad \text{(E) }120^o $[[2002 AMC 12P Problems/Problem 16|Solution]]

== Problem 17 ==

Let$ (Error compiling LaTeX. Unknown error_msg)f(x) = $\text{(A) }\frac {1}{5} \qquad \text{(B) }\frac {1}{4} \qquad \text{(C) }\frac {5}{16} \qquad \text{(D) }\frac {3}{8} \qquad \text{(E) }\frac {1}{2}$

Solution

Problem 18

A circle centered at $A$ with a radius of 1 and a circle centered at $B$ with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is

$[asy] unitsize(0.75cm); pair A=(0,1), B=(4,4); dot(A); dot(B); draw( circle(A,1) ); draw( circle(B,4) ); draw( (-1.5,0)--(8.5,0) ); draw( A -- (A+(-1,0)) ); label("$1$", A -- (A+(-1,0)), N ); draw( B -- (B+(4,0)) ); label("$4$", B -- (B+(4,0)), N ); label("$A$",A,E); label("$B$",B,W); filldraw( circle( (12/9,4/9), 4/9 ), lightgray, black ); dot( (12/9,4/9) ); [/asy]$

$\text{(A) }\frac {1}{3} \qquad \text{(B) }\frac {2}{5} \qquad \text{(C) }\frac {5}{12} \qquad \text{(D) }\frac {4}{9} \qquad \text{(E) }\frac {1}{2}$

Solution

Problem 19

The polynomial $P(x)=x^3+ax^2+bx+c$ has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the $y$ -intercept of the graph of $y=P(x)$ is 2, what is $b$ ?

$\text{(A) }-11 \qquad \text{(B) }-10 \qquad \text{(C) }-9 \qquad \text{(D) }1 \qquad \text{(E) }5$

Solution

Problem 20

Points $A = (3,9)$ , $B = (1,1)$ , $C = (5,3)$ , and $D=(a,b)$ lie in the first quadrant and are the vertices of quadrilateral $ABCD$ . The quadrilateral formed by joining the midpoints of $\overline{AB}$ , $\overline{BC}$ , $\overline{CD}$ , and $\overline{DA}$ is a square. What is the sum of the coordinates of point $D$ ?

$\text{(A) }7 \qquad \text{(B) }9 \qquad \text{(C) }10 \qquad \text{(D) }12 \qquad \text{(E) }16$

Solution

Problem 21

Four positive integers $a$ , $b$ , $c$ , and $d$ have a product of $8!$ and satisfy:

$\begin{align*} ab + a + b & = 524 \\ bc + b + c & = 146 \\ cd + c + d & = 104 \end{align*}$

What is $a-d$ ?

$\text{(A) }4 \qquad \text{(B) }6 \qquad \text{(C) }8 \qquad \text{(D) }10 \qquad \text{(E) }12$

Solution

Problem 22

In rectangle $ABCD$ , points $F$ and $G$ lie on $AB$ so that $AF=FG=GB$ and $E$ is the midpoint of $\overline{DC}$ . Also, $\overline{AC}$ intersects $\overline{EF}$ at $H$ and $\overline{EG}$ at $J$ . The area of the rectangle $ABCD$ is $70$ . Find the area of triangle $EHJ$ .

$\text{(A) }\frac {5}{2} \qquad \text{(B) }\frac {35}{12} \qquad \text{(C) }3 \qquad \text{(D) }\frac {7}{2} \qquad \text{(E) }\frac {35}{8}$

Solution

Problem 23

A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros, both of which are integers. Which of the following can also be a zero of the polynomial?

$\text{(A) }\frac {1 + i \sqrt {11}}{2} \qquad \text{(B) }\frac {1 + i}{2} \qquad \text{(C) }\frac {1}{2} + i \qquad \text{(D) }1 + \frac {i}{2} \qquad \text{(E) }\frac {1 + i \sqrt {13}}{2}$

Solution

Problem 24

In $\triangle ABC$ , $\angle ABC=45^\circ$ . Point $D$ is on $\overline{BC}$ so that $2\cdot BD=CD$ and $\angle DAB=15^\circ$ . Find $\angle ACB$ .

$\text{(A) }54^\circ \qquad \text{(B) }60^\circ \qquad \text{(C) }72^\circ \qquad \text{(D) }75^\circ \qquad \text{(E) }90^\circ$

Solution

Problem 25

Consider sequences of positive real numbers of the form $x, 2000, y, \dots$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $x$ does the term 2001 appear somewhere in the sequence?

$\text{(A) }1 \qquad \text{(B) }2 \qquad \text{(C) }3 \qquad \text{(D) }4 \qquad \text{(E) more than }4$

Solution

2001 AMC 12 (Problems • Answer Key • Resources)
Preceded by 2000 AMC 12 Problems	Followed by 2002 AMC 12A Problems
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Difference between revisions of "2002 AMC 12P Problems"

Revision as of 21:50, 29 December 2023

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See also

@@ Line 19: / Line 19: @@
 The dimensions of a rectangular box in inches are all positive integers and the volume of the box is <math>2002</math> in^3. Find the minimum possible sum of the three dimensions.
-<math>\text{(A)}\ \36\qquad \text{(B)}\ \38\qquad \text{(C)}\ \42\qquad \text{(D)}\ \44\qquad \text{(E)}\ \92</math>
+<math>
+\text{(A) }36
+\qquad
+\text{(B) }38
+\qquad
+\text{(C) }42
+\qquad
+\text{(D) }44
+\qquad
+\text{(E) }92
+</math>
 [[2002 AMC 12P Problems/Problem 3|Solution]]
@@ Line 45: / Line 55: @@
 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>\text{(A)}\ \frac{1}{3}\qquad \text{(B)}\ \frac{4}{11}\qquad \text{(C)}\ \frac{2}{5}\qquad \text{(D)}\ \frac{4}{9}\qquad \text{(E)}\ \frac{1}{2}</math>
+<math>
+\text{(A) }\frac{1}{3}
+\qquad
+\text{(B) }\frac{4}{11}
+\qquad
+\text{(C) }\frac{2}{5}
+\qquad
+\text{(D) }\frac{4}{9}
+\qquad
+\text{(E) }\frac{1}{2}
+</math>
 [[2002 AMC 12P Problems/Problem 6|Solution]]
@@ Line 53: / Line 74: @@
 How many three-digit numbers have at least one 2 and at least one 3?
-<math>\text{(A) }\52\qquad \text{(B) }\54\qquad \text{(C) }\56\qquad \text{(D) }\58\qquad \text{(E) }\60</math>
+<math>
+\text{(A) }52
+\qquad
+\text{(B) }54
+\qquad
+\text{(C) }56
+\qquad
+\text{(D) }58
+\qquad
+\text{(E) }60
+</math>
 [[2002 AMC 12P Problems/Problem 7|Solution]]
@@ Line 61: / Line 92: @@
 Let <math>AB</math> be a segment of length <math>26</math>, and let points <math>C</math> and <math>D</math> be located on <math>AB</math> such that <math>AC=1</math> and <math>AD=8</math>. Let <math>E</math> and <math>F</math> be points on one of the semicircles with diameter <math>AB</math> for which <math>EC</math> and <math>FD</math> are perpendicular to <math>AB</math>. Find <math>EF.</math>
-<math>\text{(A) }\5\qquad \text{(B) }\5\sqrt{2}\qquad \text{(C) }\7\qquad \text{(D) }\7\sqrt{2}\qquad \text{(E) }\12</math>
+<math>
+\text{(A) }5
+\qquad
+\text{(B) }5 \sqrt{2}
+\qquad
+\text{(C) }7
+\qquad
+\text{(D) }7 \sqrt{2}
+\qquad
+\text{(E) }12
+</math>
 [[2002 AMC 12P Problems/Problem 8|Solution]]
@@ Line 75: / Line 115: @@
 == Problem 10 ==
-Let <math>f_n (x) = sin^n x + cos^n x.</math> For how many <math>x</math> in <math>[0,π]</math> is it true that
+Let <math>f_n (x) = sin^n x + cos^n x.</math> For how many <math>x</math> in [<math>0,π</math>]<math> is it true that
-<math>
+</math>
 \text{(A) }2
 \qquad
@@ Line 87: / Line 127: @@
 \qquad
 \text{(E) }more than 8
-</math>
+<math>
 [[2002 AMC 12P Problems/Problem 10|Solution]]
@@ Line 93: / Line 133: @@
 == Problem 11 ==
-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}{t_2002}</cmath>
-<math>
+</math>
 \text{(A) }\frac {4003}{2003}
 \qquad
@@ Line 107: / Line 147: @@
 \qquad
 \text{(E) }2
-</math>
+<math>
 [[2002 AMC 12P Problems/Problem 11|Solution]]
@@ Line 113: / Line 153: @@
 == Problem 12 ==
-What is the maximum value of <math>n</math> for which there is a set of distinct positive integers <math>k_1, k_2, ... k_n</math> for which
+For how many positive integers </math>n<math> is </math>n^3 - 8n^2 + 20n - 13<math> a prime number?
-<math></math>k^2_1
+</math>
-<math>
+\text{(A) }one
-\text{(A) }768
 \qquad
-\text{(B) }801
+\text{(B) }two
 \qquad
-\text{(C) }934
+\text{(C) }three
 \qquad
-\text{(D) }1067
+\text{(D) }four
 \qquad
-\text{(E) }1167
+\text{(E) }more than four
-</math>
+<math>
 [[2002 AMC 12P Problems/Problem 12|Solution]]
@@ Line 132: / Line 171: @@
 == Problem 13 ==
-The parabola with equation <math>y=ax^2+bx+c</math> and vertex <math>(h,k)</math> is reflected about the line <math>y=k</math>. This results in the parabola with equation <math>y=dx^2+ex+f</math>. Which of the following equals <math>a+b+c+d+e+f</math>?
+What is the maximum value of </math>n<math> for which there is a set of distinct positive integers </math>k_1, k_2, ... k_n<math> for which
+</math>k^2_1 + k^2_2 + ... + k^2_n = 2002.<math>
-<math>
+</math>
-\text{(A) }2b
+\text{(A) }14
 \qquad
-\text{(B) }2c
+\text{(B) }15
 \qquad
-\text{(C) }2a+2b
+\text{(C) }16
 \qquad
-\text{(D) }2h
+\text{(D) }17
 \qquad
-\text{(E) }2k
+\text{(E) }18
-</math>
+<math>
 [[2002 AMC 12P Problems/Problem 13|Solution]]
@@ Line 150: / Line 191: @@
 == Problem 14 ==
-Given the nine-sided regular polygon <math>A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9</math>, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set <math>\{A_1,A_2,\dots,A_9\}</math>?
+Find </math>i + 2i^2 +3i^3 + ... + 2002i^2002.<math>
-<math>
+</math>
-\text{(A) }30
+\text{(A) }-999 + 1002i
 \qquad
-\text{(B) }36
+\text{(B) }-1002 + 999i
 \qquad
-\text{(C) }63
+\text{(C) }-1001 + 1000i
 \qquad
-\text{(D) }66
+\text{(D) }-1002 + 1001i
 \qquad
-\text{(E) }72
+\text{(E) }i
-</math>
+<math>
 [[2002 AMC 12P Problems/Problem 14|Solution]]
 == Problem 15 ==
-An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.)
+There are </math>1001 red marbles and <math>1001 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>
-<math>
+</math>
-\text{(A) }\frac {1}{2} \sqrt {3}
+\text{(A) }0
 \qquad
-\text{(B) }1
+\text{(B) }\frac{1}{2002}
 \qquad
-\text{(C) }\sqrt {2}
+\text{(C) }\frac{1}{2001}
 \qquad
-\text{(D) }\frac {3}{2}
+\text{(D) }\frac {2}{2001}
 \qquad
-\text{(E) }2
+\text{(E) }\frac{1}{1000}
-</math>
+<math>
 [[2002 AMC 12P Problems/Problem 15|Solution]]
@@ Line 185: / Line 226: @@
 == Problem 16 ==
-A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?
+The altitudes of a triangle are </math>12, 15,<math> and </math>20.<math> The largest angle in this triangle is
-<math>
+</math>
-\text{(A) }8!
+\text{(A) }72^o
 \qquad
-\text{(B) }2^8 \cdot 8!
+\text{(B) }75^o
 \qquad
-\text{(C) }(8!)^2
+\text{(C) }90^o
 \qquad
-\text{(D) }\frac {16!}{2^8}
+\text{(D) }108^o
 \qquad
-\text{(E) }16!
+\text{(E) }120^o
-</math>
+<math>
 [[2002 AMC 12P Problems/Problem 16|Solution]]
@@ Line 203: / Line 244: @@
 == Problem 17 ==
-A point <math>P</math> is selected at random from the interior of the pentagon with vertices <math>A = (0,2)</math>, <math>B = (4,0)</math>, <math>C = (2 \pi + 1, 0)</math>, <math>D = (2 \pi + 1,4)</math>, and <math>E=(0,4)</math>. What is the probability that <math>\angle APB</math> is obtuse?
+Let </math>f(x) =
 <math>
 \text{(A) }\frac {1}{5}