Difference between revisions of "2002 AMC 12P Problems"

Revision as of 21:38, 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?

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

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 $\text{(A) }1-\sqrt{2}\text{sin} x \qquad \text{(B) }-1+\sqrt{2}\text{cos} x \qquad \text{(C) }\text{cos} \frac{x}{2} - \text{sin} \frac{x}{2} \qquad \text{(D) }\text{cos} x - \text{sin} x \qquad \text{(E) }\text{cos} 2x$

Solution

Problem 18

If $a,b,c$ are real numbers such that $a^2 + 2b =7$ , $b^2 + 4c= -7,$ and $c^2 + 6a= -14$ , find $a^2 + b^2 + c^2.$

$\text{(A) }14 \qquad \text{(B) }21 \qquad \text{(C) }28 \qquad \text{(D) }35 \qquad \text{(E) }49$

Solution

Problem 19

In quadrilateral $ABCD$ , $m\angle B = m \angle C = 120^{\circ}, AB=3, BC=4, and CD=5.$ Find the area of $ABCD.$

$\text{(A) }15 \qquad \text{(B) }9 \sqrt{3} \qquad \text{(C) }\frac{45 \sqrt{3}}{4} \qquad \text{(D) }\frac{47 \sqrt{3}}{4} \qquad \text{(E) }15 \sqrt{3}$

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

@@ Line 272: / Line 272: @@
 == Problem 17 ==
-Let <math>f(x) = \sqrt{\text{sin}^4 + 4 \text{cos}^2 x} - \sqrt{\text{cos}^4 + 4 \text{sin}^2 x}.</math> An equivalent form of <math>f(x) is
+Let <math>f(x) = \sqrt{\text{sin}^4 + 4 \text{cos}^2 x} - \sqrt{\text{cos}^4 + 4 \text{sin}^2 x}.</math> An equivalent form of <math>f(x)</math> is
-</math>
+<math>
 \text{(A) }1-\sqrt{2}\text{sin} x
 \qquad
@@ Line 283: / Line 283: @@
 \qquad
 \text{(E) }\text{cos} 2x
-<math>
+</math>
 [[2002 AMC 12P Problems/Problem 17|Solution]]
@@ Line 289: / Line 289: @@
 == Problem 18 ==
-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>
+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>
-</math>
+<math>
 \text{(A) }14
 \qquad
@@ Line 301: / Line 301: @@
 \qquad
 \text{(E) }49
-<math>
+</math>
 [[2002 AMC 12P Problems/Problem 18|Solution]]
@@ Line 307: / Line 307: @@
 == 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, and CD=5.</math> Find the area of <math>ABCD.</math>
-</math>
+<math>
 \text{(A) }15
 \qquad
@@ Line 319: / Line 319: @@
 \qquad
 \text{(E) }15 \sqrt{3}
-<math>
+</math>
 [[2002 AMC 12P Problems/Problem 19|Solution]]
@@ Line 325: / Line 325: @@
 == 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
 \qquad
@@ Line 337: / Line 337: @@
 \qquad
 \text{(E) }16
-<math>
+</math>
 [[2002 AMC 12P Problems/Problem 20|Solution]]
@@ Line 343: / Line 343: @@
 == 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>
@@ Line 355: / Line 355: @@
 </cmath>
-What is </math>a-d<math>?
+What is <math>a-d</math>?
-</math>
+<math>
 \text{(A) }4
 \qquad
@@ Line 367: / Line 367: @@
 \qquad
 \text{(E) }12
-<math>
+</math>
 [[2001 AMC 12 Problems/Problem 21|Solution]]
@@ Line 373: / Line 373: @@
 == 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}
 \qquad
@@ Line 385: / Line 385: @@
 \qquad
 \text{(E) }\frac {35}{8}
-<math>
+</math>
 [[2002 AMC 12P Problems/Problem 22|Solution]]
@@ Line 393: / 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?
-</math>
+<math>
 \text{(A) }\frac {1 + i \sqrt {11}}{2}
 \qquad
@@ Line 403: / Line 403: @@
 \qquad
 \text{(E) }\frac {1 + i \sqrt {13}}{2}
-<math>
+</math>
 [[2002 AMC 12P Problems/Problem 23|Solution]]
@@ Line 409: / Line 409: @@
 == 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
 \qquad
@@ Line 421: / Line 421: @@
 \qquad
 \text{(E) }90^\circ
-<math>
+</math>
 [[2002 AMC 12P Problems/Problem 24|Solution]]
@@ Line 427: / Line 427: @@
 == 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
 \qquad
@@ Line 439: / Line 439: @@
 \qquad
 \text{(E) more than }4
-$
+</math>
 [[2002 AMC 12P Problems/Problem 25|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 21:38, 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 11

Problem 12

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

See also