2002 AMC 12P Problems

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.
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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 $f_n (x) = \text{sin}^n x + \text{cos}^n x.$ 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

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

Solution

Problem 17

Let $f(x) = \sqrt{\text{sin}^4 + 4 \text{cos}^2 x} - \sqrt{\text{cos}^4 + 4 \text{sin}^2 x}.$ An equivalent form of $f(x) is$ \text{(A) }1-\sqrt{2}\text{sin} x \qquad \text{(B) }-1+\sqrt{2}\text{cos} x \qquad \text{(C) }\text{cos} \frac{x}{2} - \text{sin} \frac{x}{2} \qquad \text{(D) }\text{cos} x - \text{sin} x \qquad \text{(E) }\text{cos} 2x $[[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

2001 AMC 12 (Problems • Answer Key • Resources)
Preceded by 2000 AMC 12 Problems	Followed by 2002 AMC 12A Problems
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All AMC 12 Problems and Solutions

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2002 AMC 12P Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

See also