Difference between revisions of "2002 AMC 10P Problems"

Latest revision as of 17:44, 20 October 2024

2002 AMC 10P (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 SAT 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

The ratio $\frac{(2^4)^8}{(4^8)^2}$ equals

$\text{(A) }\frac{1}{4} \qquad \text{(B) }\frac{1}{2} \qquad \text{(C) }1 \qquad \text{(D) }2 \qquad \text{(E) }8$

Solution

Problem 2

The sum of eleven consecutive integers is $2002.$ What is the least of these integers?

$\text{(A) }175 \qquad \text{(B) }177 \qquad \text{(C) }179 \qquad \text{(D) }180 \qquad \text{(E) }181$

Solution

Problem 3

Mary typed a six-digit number, but the two $1$ s she typed didn't show. What appeared was $2002.$ How many different six-digit numbers could she have typed?

$\text{(A) }4 \qquad \text{(B) }8 \qquad \text{(C) }10 \qquad \text{(D) }15 \qquad \text{(E) }20$

Solution

Problem 4

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 5

Let $(a_n)_{n \geq 1}$ be a sequence such that $a_1 = 1$ and $3a_{n+1} - 3a_n = 1$ for all $n \geq 1.$ Find $a_{2002}.$

$\text{(A) }666 \qquad \text{(B) }667 \qquad \text{(C) }668 \qquad \text{(D) }669 \qquad \text{(E) }670$

Solution

Problem 6

The perimeter of a rectangle is $100$ and its diagonal has length $x.$ What is the area of this rectangle? $\text{(A) }625-x^2 \qquad \text{(B) }625-\frac{x^2}{2} \qquad \text{(C) }1250-x^2 \qquad \text{(D) }1250-\frac{x^2}{2} \qquad \text{(E) }2500-\frac{x^2}{2}$

Solution

Problem 7

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 8

How many ordered triples of positive integers $(x,y,z)$ satisfy $(x^y)^z=64?$

$\text{(A) }5 \qquad \text{(B) }6 \qquad \text{(C) }7 \qquad \text{(D) }8 \qquad \text{(E) }9$

Solution

Problem 9

The function $f$ is given by the table

$\begin{tabular}{|c||c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 4 & 1 & 3 & 5 & 2 \\ \hline \end{tabular}$

If $u_0=4$ and $u_{n+1} = f(u_n)$ for $n \ge 0$ , find $u_{2002}$

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

Solution

Problem 10

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

Find $\frac{a}{b}$

$\text{(A) }0.6 \qquad \text{(B) }0.7 \qquad \text{(C) }0.8 \qquad \text{(D) }0.9 \qquad \text{(E) }1$

Solution

Problem 11

Let $P(x)=kx^3 + 2k^2x^2+k^3.$ Find the sum of all real numbers $k$ for which $x-2$ is a factor of $P(x).$

$\text{(A) }-8 \qquad \text{(B) }-4 \qquad \text{(C) }0 \qquad \text{(D) }5 \qquad \text{(E) }8$

Solution

Problem 12

For $f_n(x)=x^n$ and $a \neq 1$ consider

$\text{I. } (f_{11}(a)f_{13}(a))^{14}$

$\text{II. } f_{11}(a)f_{13}(a)f_{14}(a)$

$\text{III. } (f_{11}(f_{13}(a)))^{14}$

$\text{IV. } f_{11}(f_{13}(f_{14}(a)))$

Which of these equal $f_{2002}(a)?$

$\text{(A) I and II only} \qquad \text{(B) II and III only} \qquad \text{(C) III and IV only} \qquad \text{(D) II, III, and IV only} \qquad \text{(E) all of them}$

Solution

Problem 13

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 14

The vertex $E$ of a square $EFGH$ is at the center of square $ABCD.$ The length of a side of $ABCD$ is $1$ and the length of a side of $EFGH$ is $2.$ Side $EF$ intersects $CD$ at $I$ and $EH$ intersects $AD$ at $J.$ If angle $EID=60^{\circ},$ the area of quadrilateral $EIDJ$ is

$\text{(A) }\frac{1}{4} \qquad \text{(B) }\frac{\sqrt{3}}{6} \qquad \text{(C) }\frac{1}{3} \qquad \text{(D) }\frac{\sqrt{2}}{4} \qquad \text{(E) }\frac{\sqrt{3}}{2}$

Solution

Problem 15

What is the smallest integer $n$ for which any subset of $\{ 1, 2, 3, \; \dots \; , 20 \}$ of size $n$ must contain two numbers that differ by 8?

$\text{(A) }2 \qquad \text{(B) }8 \qquad \text{(C) }12 \qquad \text{(D) }13 \qquad \text{(E) }15$

Solution

Problem 16

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?

$\text{(A) }\sqrt{13} \qquad \text{(B) }\sqrt{14} \qquad \text{(C) }\sqrt{15} \qquad \text{(D) }4 \qquad \text{(E) }\sqrt{17}$

Solution

Problem 17

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 18

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 19

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 20

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 21

Let $f$ be a real-valued function such that

$f(x) + 2f(\frac{2002}{x}) = 3x$

for all $x>0.$ Find $f(2).$

$\text{(A) }1000 \qquad \text{(B) }2000 \qquad \text{(C) }3000 \qquad \text{(D) }4000 \qquad \text{(E) }6000$

Solution

Problem 22

In how many zeroes does the number $\frac{2002!}{(1001!)^2}$ end?

$\text{(A) }0 \qquad \text{(B) }1 \qquad \text{(C) }2 \qquad \text{(D) }200 \qquad \text{(E) }400$

Solution

Problem 23

Let $a=\frac{1^2}{1} + \frac{2^2}{3} + \frac{3^2}{5} + \; \dots \; + \frac{1001^2}{2001}$

and

$b=\frac{1^2}{3} + \frac{2^2}{5} + \frac{3^2}{7} + \; \dots \; + \frac{1001^2}{2003}.$

Find the integer closest to $a-b.$

$\text{(A) }500 \qquad \text{(B) }501 \qquad \text{(C) }999 \qquad \text{(D) }1000 \qquad \text{(E) }1001$

Solution

Problem 24

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 25

Under the new AMC $10, 12$ scoring method, $6$ points are given for each correct answer, $2.5$ points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between $0$ and $150$ can be obtained in only one way, for example, the only way to obtain a score of $146.5$ is to have $24$ correct answers and one unanswered question. Some scores can be obtained in exactly two ways, for example, a score of $104.5$ can be obtained with $17$ correct answers, $1$ unanswered question, and $7$ incorrect, and also with $12$ correct answers and $13$ unanswered questions. There are (three) scores that can be obtained in exactly three ways. What is their sum?

$\text{(A) }175 \qquad \text{(B) }179.5 \qquad \text{(C) }182 \qquad \text{(D) }188.5 \qquad \text{(E) }201$

Solution

2002 AMC 10P (Problems • Answer Key • Resources)
Preceded by 2001 AMC 10 Problems	Followed by 2002 AMC 10A Problems
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Difference between revisions of "2002 AMC 10P Problems"

Latest revision as of 17:44, 20 October 2024

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

@@ Line 18: / Line 18: @@
 == Problem 2 ==
-The sum of eleven consecutive integers is <math>2002.</math> What is the smallest of these integers?
+The sum of eleven consecutive integers is <math>2002.</math> What is the least of these integers?
 <math>
@@ Line 76: / Line 76: @@
 == Problem 6 ==
-The perimeter of a rectangle <math>100</math> and its diagonal has length <math>x.</math> What is the area of this rectangle?
+The perimeter of a rectangle is <math>100</math> and its diagonal has length <math>x.</math> What is the area of this rectangle?
 <math>
 \text{(A) }625-x^2
@@ Line 154: / Line 154: @@
 </math>
-[[2002 AMC 10P Problems/Problem 2|Solution]]
+[[2002 AMC 10P Problems/Problem 9|Solution]]
 == Problem 10 ==
@@ Line 164: / Line 164: @@
 <math>
-\text{(A) }0.4
+\text{(A) }0.6
 \qquad
-\text{(B) }0.5
+\text{(B) }0.7
 \qquad
-\text{(C) }0.6
+\text{(C) }0.8
 \qquad
-\text{(D) }0.7
+\text{(D) }0.9
 \qquad
-\text{(E) }0.8
+\text{(E) }1
 </math>
@@ Line 262: / Line 262: @@
 What is the smallest integer <math>n</math> for which any subset of <math>\{ 1, 2, 3, \; \dots \; , 20 \}</math> of size <math>n</math> must contain two numbers that differ by 8?
 <math>
-\text{(A) }0
+\text{(A) }2
 \qquad
-\text{(B) }\frac{1}{2002}
+\text{(B) }8
 \qquad
-\text{(C) }\frac{1}{2001}
+\text{(C) }12
 \qquad
-\text{(D) }\frac {2}{2001}
+\text{(D) }13
 \qquad
-\text{(E) }\frac{1}{1000}
+\text{(E) }15
 </math>
@@ Line 280: / Line 279: @@
 == Problem 16 ==
-The altitudes of a triangle are <math>12, 15,</math> and <math>20.</math> The largest angle in this triangle is
+Two walls and the ceiling of a room meet at right angles at point <math>P.</math>  A fly is in the air one meter from one wall, eight meters from the other wall, and nine meters from point <math>P</math>. How many meters is the fly from the ceiling?
 <math>
-\text{(A) }72^\circ
+\text{(A) }\sqrt{13}
 \qquad
-\text{(B) }75^\circ
+\text{(B) }\sqrt{14}
 \qquad
-\text{(C) }90^\circ
+\text{(C) }\sqrt{15}
 \qquad
-\text{(D) }108^\circ
+\text{(D) }4
 \qquad
-\text{(E) }120^\circ
+\text{(E) }\sqrt{17}
 </math>
@@ Line 298: / Line 297: @@
 == Problem 17 ==
-Let <math>f(x) = \sqrt{\sin^4{x} + 4 \cos^2{x}} - \sqrt{\cos^4{x} + 4 \sin^2{x}}.</math> An equivalent form of <math>f(x)</math> is
+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>
 <math>
-\text{(A) }1-\sqrt{2}\sin{x}
+\text{(A) }0
 \qquad
-\text{(B) }-1+\sqrt{2}\cos{x}
+\text{(B) }\frac{1}{2002}
 \qquad
-\text{(C) }\cos{\frac{x}{2}} - \sin{\frac{x}{2}}
+\text{(C) }\frac{1}{2001}
 \qquad
-\text{(D) }\cos{x} - \sin{x}
+\text{(D) }\frac {2}{2001}
 \qquad
-\text{(E) }\cos{2x}
+\text{(E) }\frac{1}{1000}
 </math>
@@ Line 316: / Line 315: @@
 == 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>
+For how many positive integers <math>n</math> is <math>n^3 - 8n^2 + 20n - 13</math> a prime number?
 <math>
-\text{(A) }14
+\text{(A) one}
 \qquad
-\text{(B) }21
+\text{(B) two}
 \qquad
-\text{(C) }28
+\text{(C) three}
 \qquad
-\text{(D) }35
+\text{(D) four}
 \qquad
-\text{(E) }49
+\text{(E) more than four}
 </math>
@@ Line 334: / Line 333: @@
 == Problem 19 ==
-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>
+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>
-\text{(A) }15
+\text{(A) }14
 \qquad
-\text{(B) }9 \sqrt{3}
+\text{(B) }21
 \qquad
-\text{(C) }\frac{45 \sqrt{3}}{4}
+\text{(C) }28
 \qquad
-\text{(D) }\frac{47 \sqrt{3}}{4}
+\text{(D) }35
 \qquad
-\text{(E) }15 \sqrt{3}
+\text{(E) }49
 </math>
@@ Line 352: / Line 351: @@
 == Problem 20 ==
-Let <math>f</math> be a real-valued function such that
+How many three-digit numbers have at least one <math>2</math> and at least one <math>3</math>?
-<cmath>f(x) + 2f(\frac{2002}{x}) = 3x</cmath>
-for all <math>x>0.</math> Find <math>f(2).</math>
 <math>
-\text{(A) }1000
+\text{(A) }52
 \qquad
-\text{(B) }2000
+\text{(B) }54
 \qquad
-\text{(C) }3000
+\text{(C) }56
 \qquad
-\text{(D) }4000
+\text{(D) }58
 \qquad
-\text{(E) }6000
+\text{(E) }60
 </math>
@@ Line 374: / Line 369: @@
 == Problem 21 ==
-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>f</math> be a real-valued function such that
-<cmath>2(\log_a{c} + \log_b{c}) = 9\log_{ab}{c}.</cmath>
+<cmath>f(x) + 2f(\frac{2002}{x}) = 3x</cmath>
-Find the largest possible value of <math>\log_a b.</math>
+for all <math>x>0.</math> Find <math>f(2).</math>
 <math>
-\text{(A) }\sqrt{2}
+\text{(A) }1000
 \qquad
-\text{(B) }\sqrt{3}
+\text{(B) }2000
 \qquad
-\text{(C) }2
+\text{(C) }3000
 \qquad
-\text{(D) }\sqrt{6}
+\text{(D) }4000
 \qquad
-\text{(E) }3
+\text{(E) }6000
 </math>
@@ Line 396: / Line 391: @@
 == Problem 22 ==
-Under the new AMC <math>10, 12</math> scoring method, <math>6</math> poitns 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?
+In how many zeroes does the number <math>\frac{2002!}{(1001!)^2}</math> end?
 <math>
-\text{(A) }175
+\text{(A) }0
 \qquad
-\text{(B) }179.5
+\text{(B) }1
 \qquad
-\text{(C) }182
+\text{(C) }2
 \qquad
-\text{(D) }188.5
+\text{(D) }200
 \qquad
-\text{(E) }201
+\text{(E) }400
 </math>
@@ Line 415: / Line 409: @@
 == Problem 23 ==
-The equation <math>z(z+i)(z+3i)=2002i</math> has a zero of the form <math>a+bi</math>, where <math>a</math> and <math>b</math> are positive real numbers. Find <math>a.</math>
+Let
+<cmath>a=\frac{1^2}{1} + \frac{2^2}{3} + \frac{3^2}{5} +  \; \dots \; + \frac{1001^2}{2001}</cmath>
+and
+<cmath>b=\frac{1^2}{3} + \frac{2^2}{5} + \frac{3^2}{7} +  \; \dots \; + \frac{1001^2}{2003}.</cmath>
+Find the integer closest to <math>a-b.</math>
 <math>
-\text{(A) }\sqrt{118}
+\text{(A) }500
 \qquad
-\text{(B) }\sqrt{210}
+\text{(B) }501
 \qquad
-\text{(C) }2 \sqrt{210}
+\text{(C) }999
 \qquad
-\text{(D) }\sqrt{2002}
+\text{(D) }1000
 \qquad
-\text{(E) }100 \sqrt{2}
+\text{(E) }1001
 </math>
@@ Line 433: / Line 434: @@
 == Problem 24 ==
-Let <math>ABCD</math> be a regular tetrahedron and Let <math>E</math> be a point inside the face <math>ABC.</math> Denote by <math>s</math> the sum of the distances from <math>E</math> to the faces <math>DAB, DBC, DCA,</math> and by <math>S</math> the sum of the distances from <math>E</math> to the edges <math>AB, BC, CA.</math> Then <math>\frac{s}{S}</math> equals
+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
+<cmath>k^2_1 + k^2_2 + ... + k^2_n = 2002?</cmath>
 <math>
-\text{(A) }\sqrt{2}
+\text{(A) }14
 \qquad
-\text{(B) }\frac{2 \sqrt{2}}{3}
+\text{(B) }15
 \qquad
-\text{(C) }\frac{\sqrt{6}}{2}
+\text{(C) }16
 \qquad
-\text{(D) }2
+\text{(D) }17
 \qquad
-\text{(E) }3
+\text{(E) }18
 </math>
@@ Line 451: / Line 454: @@
 == Problem 25 ==
-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>
+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>
-\text{(A) }\frac{1}{2}
+\text{(A) }175
 \qquad
-\text{(B) }\frac{\sqrt{2}}{2}
+\text{(B) }179.5
 \qquad
-\text{(C) }\frac{\sqrt{3}}{2}
+\text{(C) }182
 \qquad
-\text{(D) }\frac{\sqrt{6}}{2}
+\text{(D) }188.5
 \qquad
-\text{(E) }1
+\text{(E) }201
 </math>