Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2002 AMC 12P Problems"
Line 143: Line 143:
 
== Problem 10 ==
 
== 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,\pi]</math> is it true that
  
 
<math>
 
<math>
Line 184: Line 184:
  
 
<math>
 
<math>
−
\text{(A) }one
+
\text{(A) one}
 
\qquad
 
\qquad
−
\text{(B) }two
+
\text{(B) two}
 
\qquad
 
\qquad
−
\text{(C) }three
+
\text{(C) three}
 
\qquad
 
\qquad
−
\text{(D) }four
+
\text{(D) four}
 
\qquad
 
\qquad
−
\text{(E) }more than four
+
\text{(E) more than four}
 
</math>
 
</math>
  
Line 289: Line 289:
 
== Problem 18 ==
 
== Problem 18 ==
  
−
A circle centered at </math>A<math> with a radius of 1 and a circle centered at </math>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
+
If </math>a,b,c<math> are real numbers such that </math>a^2 + 2b =7<math>, </math>b^2 + 4c= -7,<math> and </math>c^2 + 6a= -14<math>, find </math>a^2 + b^2 + c^2.<math>
  
−
<asy>
+
</math>
−
unitsize(0.75cm);
+
\text{(A) }14
−
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>
 
−
 
 
−
<math>
 
−
\text{(A) }\frac {1}{3}
 
 
\qquad
 
\qquad
−
\text{(B) }\frac {2}{5}
+
\text{(B) }21
 
\qquad
 
\qquad
−
\text{(C) }\frac {5}{12}
+
\text{(C) }28
 
\qquad
 
\qquad
−
\text{(D) }\frac {4}{9}
+
\text{(D) }35
 
\qquad
 
\qquad
−
\text{(E) }\frac {1}{2}
+
\text{(E) }49
−
</math>
+
<math>
  
 
[[2002 AMC 12P Problems/Problem 18|Solution]]
 
[[2002 AMC 12P Problems/Problem 18|Solution]]
Line 325: Line 307:
 
== Problem 19 ==
 
== Problem 19 ==
  
−
The polynomial <math>P(x)=x^3+ax^2+bx+c</math> 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 <math>y</math>-intercept of the graph of <math>y=P(x)</math> is 2, what is <math>b</math>?
+
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>
  
−
<math>
+
</math>
−
\text{(A) }-11
+
\text{(A) }15
 
\qquad
 
\qquad
−
\text{(B) }-10
+
\text{(B) }9 \sqrt{3}
 
\qquad
 
\qquad
−
\text{(C) }-9
+
\text{(C) }\frac{45 \sqrt{3}}{4}
 
\qquad
 
\qquad
−
\text{(D) }1
+
\text{(D) }\frac{47 \sqrt{3}}{4}
 
\qquad
 
\qquad
−
\text{(E) }5
+
\text{(E) }15 \sqrt{3}
−
</math>
+
<math>
  
 
[[2002 AMC 12P Problems/Problem 19|Solution]]
 
[[2002 AMC 12P Problems/Problem 19|Solution]]
Line 343: Line 325:
 
== Problem 20 ==
 
== Problem 20 ==
  
−
Points <math>A = (3,9)</math>, <math>B = (1,1)</math>, <math>C = (5,3)</math>, and <math>D=(a,b)</math> lie in the first quadrant and are the vertices of quadrilateral <math>ABCD</math>. The quadrilateral formed by joining the midpoints of <math>\overline{AB}</math>, <math>\overline{BC}</math>, <math>\overline{CD}</math>, and <math>\overline{DA}</math> is a square. What is the sum of the coordinates of point <math>D</math>?
+
Points </math>A = (3,9)<math>, </math>B = (1,1)<math>, </math>C = (5,3)<math>, and </math>D=(a,b)<math> lie in the first quadrant and are the vertices of quadrilateral </math>ABCD<math>. The quadrilateral formed by joining the midpoints of </math>\overline{AB}<math>, </math>\overline{BC}<math>, </math>\overline{CD}<math>, and </math>\overline{DA}<math> is a square. What is the sum of the coordinates of point </math>D<math>?
  
−
<math>
+
</math>
 
\text{(A) }7
 
\text{(A) }7
 
\qquad
 
\qquad
Line 355: Line 337:
 
\qquad
 
\qquad
 
\text{(E) }16
 
\text{(E) }16
−
</math>
+
<math>
  
 
[[2002 AMC 12P Problems/Problem 20|Solution]]
 
[[2002 AMC 12P Problems/Problem 20|Solution]]
Line 361: Line 343:
 
== Problem 21 ==
 
== Problem 21 ==
  
−
Four positive integers <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> have a product of <math>8!</math> and satisfy:
+
Four positive integers </math>a<math>, </math>b<math>, </math>c<math>, and </math>d<math> have a product of </math>8!<math> and satisfy:
  
 
<cmath>
 
<cmath>
Line 373: Line 355:
 
</cmath>
 
</cmath>
  
−
What is <math>a-d</math>?
+
What is </math>a-d<math>?
  
−
<math>
+
</math>
 
\text{(A) }4
 
\text{(A) }4
 
\qquad
 
\qquad
Line 385: Line 367:
 
\qquad
 
\qquad
 
\text{(E) }12
 
\text{(E) }12
−
</math>
+
<math>
  
 
[[2001 AMC 12 Problems/Problem 21|Solution]]
 
[[2001 AMC 12 Problems/Problem 21|Solution]]
Line 391: Line 373:
 
== Problem 22 ==
 
== Problem 22 ==
  
−
In rectangle <math>ABCD</math>, points <math>F</math> and <math>G</math> lie on <math>AB</math> so that <math>AF=FG=GB</math> and <math>E</math> is the midpoint of <math>\overline{DC}</math>. Also, <math>\overline{AC}</math> intersects <math>\overline{EF}</math> at <math>H</math> and <math>\overline{EG}</math> at <math>J</math>. The area of the rectangle <math>ABCD</math> is <math>70</math>. Find the area of triangle <math>EHJ</math>.
+
In rectangle </math>ABCD<math>, points </math>F<math> and </math>G<math> lie on </math>AB<math> so that </math>AF=FG=GB<math> and </math>E<math> is the midpoint of </math>\overline{DC}<math>. Also, </math>\overline{AC}<math> intersects </math>\overline{EF}<math> at </math>H<math> and </math>\overline{EG}<math> at </math>J<math>. The area of the rectangle </math>ABCD<math> is </math>70<math>. Find the area of triangle </math>EHJ<math>.
  
−
<math>
+
</math>
 
\text{(A) }\frac {5}{2}
 
\text{(A) }\frac {5}{2}
 
\qquad
 
\qquad
Line 403: Line 385:
 
\qquad
 
\qquad
 
\text{(E) }\frac {35}{8}
 
\text{(E) }\frac {35}{8}
−
</math>
+
<math>
  
 
[[2002 AMC 12P Problems/Problem 22|Solution]]
 
[[2002 AMC 12P Problems/Problem 22|Solution]]
Line 411: Line 393:
 
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?
 
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?
  
−
<math>
+
</math>
 
\text{(A) }\frac {1 + i \sqrt {11}}{2}
 
\text{(A) }\frac {1 + i \sqrt {11}}{2}
 
\qquad
 
\qquad
Line 421: Line 403:
 
\qquad
 
\qquad
 
\text{(E) }\frac {1 + i \sqrt {13}}{2}
 
\text{(E) }\frac {1 + i \sqrt {13}}{2}
−
</math>
+
<math>
  
 
[[2002 AMC 12P Problems/Problem 23|Solution]]
 
[[2002 AMC 12P Problems/Problem 23|Solution]]
Line 427: Line 409:
 
== Problem 24 ==
 
== Problem 24 ==
  
−
In <math>\triangle ABC</math>, <math>\angle ABC=45^\circ</math>. Point <math>D</math> is on <math>\overline{BC}</math> so that <math>2\cdot BD=CD</math> and <math>\angle DAB=15^\circ</math>. Find <math>\angle ACB</math>.
+
In </math>\triangle ABC<math>, </math>\angle ABC=45^\circ<math>. Point </math>D<math> is on </math>\overline{BC}<math> so that </math>2\cdot BD=CD<math> and </math>\angle DAB=15^\circ<math>. Find </math>\angle ACB<math>.
  
−
<math>
+
</math>
 
\text{(A) }54^\circ
 
\text{(A) }54^\circ
 
\qquad
 
\qquad
Line 439: Line 421:
 
\qquad
 
\qquad
 
\text{(E) }90^\circ
 
\text{(E) }90^\circ
−
</math>
+
<math>
  
 
[[2002 AMC 12P Problems/Problem 24|Solution]]
 
[[2002 AMC 12P Problems/Problem 24|Solution]]
Line 445: Line 427:
 
== Problem 25 ==
 
== Problem 25 ==
  
−
Consider sequences of positive real numbers of the form <math>x, 2000, y, \dots</math> in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of <math>x</math> does the term 2001 appear somewhere in the sequence?
+
Consider sequences of positive real numbers of the form </math>x, 2000, y, \dots<math> in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of </math>x<math> does the term 2001 appear somewhere in the sequence?
  
−
<math>
+
</math>
 
\text{(A) }1
 
\text{(A) }1
 
\qquad
 
\qquad
Line 457: Line 439:
 
\qquad
 
\qquad
 
\text{(E) more than }4
 
\text{(E) more than }4
−
</math>
+
$
  
 
[[2002 AMC 12P Problems/Problem 25|Solution]]
 
[[2002 AMC 12P Problems/Problem 25|Solution]]

Revision as of 21:25, 29 December 2023

2002 AMC 12P (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. 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.
  2. 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.
  3. 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).
  4. Figures are not necessarily drawn to scale.
  5. 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?

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

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 \[\frac{2002}{m^2 -2}\]

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

Solution

Problem 6

Participation in the local soccer league this year is $10$% higher than last year. The number of males increased by $5$% and the number of females increased by $20$%. 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 $[0,\pi]$ is it true that

$\text{(A) }2 \qquad \text{(B) }4 \qquad \text{(C) }6 \qquad \text{(D) }8 \qquad \text{(E) }more than 8$

Solution

Problem 11

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

\[\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + ... + \frac{1}{t_2002}\]

$\text{(A) }\frac {4003}{2003} \qquad \text{(B) }\frac {2001}{1001} \qquad \text{(C) }\frac {4004}{2003} \qquad \text{(D) }\frac {4001}{2001} \qquad \text{(E) }2$

Solution

Problem 12

For how many positive integers $n$ is $n^3 - 8n^2 + 20n - 13$ a prime number?

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

Solution

Problem 13

What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1, k_2, ... k_n$ for which

\[k^2_1 + k^2_2 + ... + k^2_n = 2002?\]

$\text{(A) }14 \qquad \text{(B) }15 \qquad \text{(C) }16 \qquad \text{(D) }17 \qquad \text{(E) }18$

Solution

Problem 14

Find $i + 2i^2 +3i^3 + ... + 2002i^{2002}.$

$\text{(A) }-999 + 1002i \qquad \text{(B) }-1002 + 999i \qquad \text{(C) }-1001 + 1000i \qquad \text{(D) }-1002 + 1001i \qquad \text{(E) }i$

Solution

Problem 15

There are $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 $|P_s-P_d|.$

$\text{(A) }0 \qquad \text{(B) }\frac{1}{2002} \qquad \text{(C) }\frac{1}{2001} \qquad \text{(D) }\frac {2}{2001} \qquad \text{(E) }\frac{1}{1000}$

Solution

Problem 16

The altitudes of a triangle are $12, 15,$ and $20.$ The largest angle in this triangle is

$\text{(A) }72^o \qquad \text{(B) }75^o \qquad \text{(C) }90^o \qquad \text{(D) }108^o \qquad \text{(E) }120^o$

Solution

Problem 17

Let $f(x) =$ \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} $[[2002 AMC 12P Problems/Problem 17|Solution]]

== Problem 18 ==

If$ (Error compiling LaTeX. Unknown error_msg)a,b,c$are real numbers such that$a^2 + 2b =7$,$b^2 + 4c= -7,$and$c^2 + 6a= -14$, find$a^2 + b^2 + c^2.$$ (Error compiling LaTeX. Unknown error_msg) \text{(A) }14 \qquad \text{(B) }21 \qquad \text{(C) }28 \qquad \text{(D) }35 \qquad \text{(E) }49 $[[2002 AMC 12P Problems/Problem 18|Solution]]

== Problem 19 ==

In quadrilateral$ (Error compiling LaTeX. Unknown error_msg)ABCD$,$m\angle B = m \angle C = 120^{\circ}, AB=3, BC=4, and CD=5.$Find the area of$ABCD.$$ (Error compiling LaTeX. Unknown error_msg) \text{(A) }15 \qquad \text{(B) }9 \sqrt{3} \qquad \text{(C) }\frac{45 \sqrt{3}}{4} \qquad \text{(D) }\frac{47 \sqrt{3}}{4} \qquad \text{(E) }15 \sqrt{3} $[[2002 AMC 12P Problems/Problem 19|Solution]]

== Problem 20 ==

Points$ (Error compiling LaTeX. Unknown error_msg)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 $[[2002 AMC 12P Problems/Problem 20|Solution]]

== Problem 21 ==

Four positive integers$ (Error compiling LaTeX. Unknown error_msg)a$,$b$,$c$, and$d$have a product of$8!$and satisfy:

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

What is$ (Error compiling LaTeX. Unknown error_msg)a-d$?$ \text{(A) }4 \qquad \text{(B) }6 \qquad \text{(C) }8 \qquad \text{(D) }10 \qquad \text{(E) }12 $[[2001 AMC 12 Problems/Problem 21|Solution]]

== Problem 22 ==

In rectangle$ (Error compiling LaTeX. Unknown error_msg)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} $[[2002 AMC 12P Problems/Problem 22|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?$ (Error compiling LaTeX. Unknown error_msg) \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} $[[2002 AMC 12P Problems/Problem 23|Solution]]

== Problem 24 ==

In$ (Error compiling LaTeX. Unknown error_msg)\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 $[[2002 AMC 12P Problems/Problem 24|Solution]]

== Problem 25 ==

Consider sequences of positive real numbers of the form$ (Error compiling LaTeX. Unknown error_msg)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

See also

2001 AMC 12 (Problems • Answer Key • Resources)
Preceded by
2000 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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2002_AMC_12P_Problems&oldid=208728"