Difference between revisions of "2002 AMC 12P Problems"

Latest revision as of 03:03, 14 July 2024

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

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

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 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 $\frac{2002}{m^2 -2}$

a positive integer?

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

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?

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

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

$6f_{4}(x)-4f_{6}(x)=2f_{2}(x)?$

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

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 $f(x) = \sqrt{\sin^4{x} + 4 \cos^2{x}} - \sqrt{\cos^4{x} + 4 \sin^2{x}}.$ An equivalent form of $f(x)$ is

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

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

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 21

Let $a$ and $b$ be real numbers greater than $1$ for which there exists a positive real number $c,$ different from $1$ , such that

$2(\log_a{c} + \log_b{c}) = 9\log_{ab}{c}.$

Find the largest possible value of $\log_a b.$

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

Solution

Problem 22

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

Problem 23

The equation $z(z+i)(z+3i)=2002i$ has a zero of the form $a+bi$ , where $a$ and $b$ are positive real numbers. Find $a.$

$\text{(A) }\sqrt{118} \qquad \text{(B) }\sqrt{210} \qquad \text{(C) }2 \sqrt{210} \qquad \text{(D) }\sqrt{2002} \qquad \text{(E) }100 \sqrt{2}$

Solution

Problem 24

Let $ABCD$ be a regular tetrahedron and Let $E$ be a point inside the face $ABC.$ Denote by $s$ the sum of the distances from $E$ to the faces $DAB, DBC, DCA,$ and by $S$ the sum of the distances from $E$ to the edges $AB, BC, CA.$ Then $\frac{s}{S}$ equals

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

Solution

Problem 25

Let $a$ and $b$ be real numbers such that $\sin{a} + \sin{b} = \frac{\sqrt{2}}{2}$ and $\cos {a} + \cos {b} = \frac{\sqrt{6}}{2}.$ Find $\sin{(a+b)}.$

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

Solution

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

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Difference between revisions of "2002 AMC 12P Problems"

Latest revision as of 03:03, 14 July 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 1: / Line 1: @@
-{{AMC12 Problems|year=2001|ab=}}
+{{AMC12 Problems|year=2002|ab=P}}
 == Problem 1 ==
-The sum of two numbers is <math>S</math>. Suppose <math>3</math> is added to each number and then
-each of the resulting numbers is doubled. What is the sum of the final two
-numbers?
-<math>\text{(A)}\ 2S + 3\qquad \text{(B)}\ 3S + 2\qquad \text{(C)}\ 3S + 6 \qquad\text{(D)} 2S + 6 \qquad \text{(E)}\ 2S + 12</math>
+Which of the following numbers is a perfect square?
-[[2001 AMC 12 Problems/Problem 1|Solution]]
+<math>
+\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
+</math>
+[[2002 AMC 12P Problems/Problem 1|Solution]]
 == Problem 2 ==
-Let <math>P(n)</math> and <math>S(n)</math> denote the product and the sum, respectively, of the digits
-of the integer <math>n</math>. For example, <math>P(23) = 6</math> and <math>S(23) = 5</math>. Suppose <math>N</math> is a
-two-digit number such that <math>N = P(N)+S(N)</math>. What is the units digit of <math>N</math>?
-<math>\text{(A)}\ 2\qquad \text{(B)}\ 3\qquad \text{(C)}\ 6\qquad \text{(D)}\ 8\qquad \text{(E)}\ 9</math>
+The function <math>f</math> is given by the table
-[[2001 AMC 12 Problems/Problem 2|Solution]]
+<cmath>
+\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}
+</cmath>
+If <math>u_0=4</math> and <math>u_{n+1} = f(u_n)</math> for <math>n \ge 0</math>, find <math>u_{2002}</math>
+<math>
+\text{(A) }1
+\qquad
+\text{(B) }2
+\qquad
+\text{(C) }3
+\qquad
+\text{(D) }4
+\qquad
+\text{(E) }5
+</math>
+[[2002 AMC 12P Problems/Problem 2|Solution]]
 == Problem 3 ==
-The state income tax where Kristin lives is levied at the rate of <math>p\%</math> of the first
-<math>\textdollar 28000</math> of annual income plus <math>(p + 2)\%</math> of any amount above <math>\textdollar 28000</math>. Kristin
-noticed that the state income tax she paid amounted to <math>(p + 0.25)\%</math> of her
-annual income. What was her annual income?
-<math>\text{(A)}\ \$28000\qquad \text{(B)}\ \$32000\qquad \text{(C)}\ \$35000\qquad \text{(D)}\ \$42000\qquad \text{(E)}\ \$56000</math>
+The dimensions of a rectangular box in inches are all positive integers and the volume of the box is <math>2002</math> in<math>^3</math>. Find the minimum possible sum of the three dimensions.
-[[2001 AMC 12 Problems/Problem 3|Solution]]
+<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]]
 == Problem 4 ==
-The mean of three numbers is <math>10</math> more than the least of the numbers and <math>15</math>
-less than the greatest. The median of the three numbers is <math>5</math>. What is their
-sum?
-<math>\text{(A)}\ 5\qquad \text{(B)}\ 20\qquad \text{(C)}\ 25\qquad \text{(D)}\ 30\qquad \text{(E)}\ 36</math>
+Let <math>a</math> and <math>b</math> be distinct real numbers for which
+<cmath>\frac{a}{b} + \frac{a+10b}{b+10a} = 2.</cmath>
-[[2001 AMC 12 Problems/Problem 4|Solution]]
+Find <math>\frac{a}{b}</math>
-== Problem 5 ==
+<math>
-What is the product of all positive odd integers less than 10000?
+\text{(A) }0.4
+\qquad
+\text{(B) }0.5
+\qquad
+\text{(C) }0.6
+\qquad
+\text{(D) }0.7
+\qquad
+\text{(E) }0.8
+</math>
-<math>\text{(A)}\ \dfrac{10000!}{(5000!)^2}\qquad \text{(B)}\ \dfrac{10000!}{2^{5000}}\qquad
+[[2002 AMC 12P Problems/Problem 4|Solution]]
-\text{(C)}\ \dfrac{9999!}{2^{5000}}\qquad \text{(D)}\ \dfrac{10000!}{2^{5000} \cdot 5000!}\qquad
-\text{(E)}\ \dfrac{5000!}{2^{5000}}</math>
-[[2001 AMC 12 Problems/Problem 5|Solution]]
+== Problem 5 ==
-== Problem 6 ==
+For how many positive integers <math>m</math> is
-A telephone number has the form <math>\text{ABC-DEF-GHIJ}</math>, where each letter represents
+<cmath>\frac{2002}{m^2 -2}</cmath>
-a different digit. The digits in each part of the number are in decreasing
-order; that is, <math>A > B > C</math>, <math>D > E > F</math>, and <math>G > H > I > J</math>. Furthermore,
-<math>D</math>, <math>E</math>, and <math>F</math> are consecutive even digits; <math>G</math>, <math>H</math>, <math>I</math>, and <math>J</math> are consecutive odd
-digits; and <math>A + B + C = 9</math>. Find <math>A</math>.
-<math>\text{(A)}\ 4\qquad \text{(B)}\ 5\qquad \text{(C)}\ 6\qquad \text{(D)}\ 7\qquad \text{(E)}\ 8</math>
+a positive integer?
-[[2001 AMC 12 Problems/Problem 6|Solution]]
+<math>
+\text{(A) one}
+\qquad
+\text{(B) two}
+\qquad
+\text{(C) three}
+\qquad
+\text{(D) four}
+\qquad
+\text{(E) more than four}
+</math>
-== Problem 7 ==
+[[2002 AMC 12P Problems/Problem 5|Solution]]
-A charity sells <math>140</math> benefit tickets for a total of <math>\textdollar 2001</math>. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?
+== Problem 6 ==
-<math>\text{(A) }\$782\qquad \text{(B) }\$986\qquad \text{(C) }\$1158\qquad \text{(D) }\$1219\qquad \text{(E) }\$1449</math>
+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?
-[[2001 AMC 12 Problems/Problem 7|Solution]]
+<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>
-== Problem 8 ==
+[[2002 AMC 12P Problems/Problem 6|Solution]]
-Which of the cones listed below can be formed from a <math>252^\circ</math> sector of a circle of radius <math>10</math> by aligning the two straight sides?
+== Problem 7 ==
-<asy>
+How many three-digit numbers have at least one <math>2</math> and at least one <math>3</math>?
-import graph;
-unitsize(1.5cm);
-defaultpen(fontsize(8pt));
-draw(Arc((0,0),1,-72,180),linewidth(.8pt));
+<math>
-draw(dir(288)--(0,0)--(-1,0),linewidth(.8pt));
+\text{(A) }52
-label("$10$",(-0.5,0),S);
+\qquad
-draw(Arc((0,0),0.1,-72,180));
+\text{(B) }54
-label("$252^{\circ}$",(0.05,0.05),NE);
+\qquad
-</asy>
+\text{(C) }56
+\qquad
+\text{(D) }58
+\qquad
+\text{(E) }60
+</math>
-<math>\text{(A) A cone with slant height of } 10 \text{ and radius } 6</math>
+[[2002 AMC 12P Problems/Problem 7|Solution]]
-<math>\text{(B) A cone with height of } 10 \text{ and radius } 6</math>
+== Problem 8 ==
-<math>\text{(C) A cone with slant height of } 10 \text{ and radius } 7</math>
+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{(D) A cone with height of } 10 \text{ and radius } 7</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{(E) A cone with slant height of } 10 \text{ and radius } 8</math>
+[[2002 AMC 12P Problems/Problem 8|Solution]]
-[[2001 AMC 12 Problems/Problem 8|Solution]]
 == Problem 9 ==
-Let <math>f</math> be a function satisfying <math>f(xy) = \frac{f(x)}y</math> for all positive real numbers <math>x</math> and <math>y</math>. If <math>f(500) =3</math>, what is the value of <math>f(600)</math>?
+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)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ \frac52 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ \frac{18}5</math>
+<math>
+\text{(A) }\sqrt{13}
+\qquad
+\text{(B) }\sqrt{14}
+\qquad
+\text{(C) }\sqrt{15}
+\qquad
+\text{(D) }4
+\qquad
+\text{(E) }\sqrt{17}
+</math>
-[[2001 AMC 12 Problems/Problem 9|Solution]]
+[[2002 AMC 12P Problems/Problem 9|Solution]]
 == Problem 10 ==
-The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
+Let <math>f_n (x) = \text{sin}^n x + \text{cos}^n x.</math> For how many <math>x</math> in <math>[0,\pi]</math> is it true that
+<cmath>6f_{4}(x)-4f_{6}(x)=2f_{2}(x)?</cmath>
 <math>
-\text{(A) }50
+\text{(A) }2
 \qquad
-\text{(B) }52
+\text{(B) }4
 \qquad
-\text{(C) }54
+\text{(C) }6
 \qquad
-\text{(D) }56
+\text{(D) }8
 \qquad
-\text{(E) }58
+\text{(E) more than }8
 </math>
-<asy>
+[[2002 AMC 12P Problems/Problem 10|Solution]]
-unitsize(3mm);
-defaultpen(linewidth(0.8pt));
-path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0);
+== Problem 11 ==
-path p2=(0,1)--(1,1)--(1,0);
-path p3=(2,0)--(2,1)--(3,1);
-path p4=(3,2)--(2,2)--(2,3);
-path p5=(1,3)--(1,2)--(0,2);
-path p6=(1,1)--(2,2);
-path p7=(2,1)--(1,2);
-path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7;
-for(int i=0; i<3; ++i)
-{
-for(int j=0; j<3; ++j)
-{
-draw(shift(3*i,3*j)*p);
-}
-}
-</asy>
-[[2001 AMC 12 Problems/Problem 10|Solution]]
-== Problem 11 ==
+Let <math>t_n = \frac{n(n+1)}{2}</math> be the <math>n</math>th triangular number. Find
-A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?
+<cmath>\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + ... + \frac{1}{t_{2002}}</cmath>
 <math>
-\text{(A) }\frac {3}{10}
+\text{(A) }\frac {4003}{2003}
 \qquad
-\text{(B) }\frac {2}{5}
+\text{(B) }\frac {2001}{1001}
 \qquad
-\text{(C) }\frac {1}{2}
+\text{(C) }\frac {4004}{2003}
 \qquad
-\text{(D) }\frac {3}{5}
+\text{(D) }\frac {4001}{2001}
 \qquad
-\text{(E) }\frac {7}{10}
+\text{(E) }2
 </math>
-[[2001 AMC 12 Problems/Problem 11|Solution]]
+[[2002 AMC 12P Problems/Problem 11|Solution]]
 == Problem 12 ==
-How many positive integers not exceeding <math>2001</math> are multiples of <math>3</math> or <math>4</math> but not <math>5</math>?
+For how many positive integers <math>n</math> is <math>n^3 - 8n^2 + 20n - 13</math> a prime number?
 <math>
-\text{(A) }768
+\text{(A) one}
 \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>
-[[2001 AMC 12 Problems/Problem 12|Solution]]
+[[2002 AMC 12P Problems/Problem 12|Solution]]
 == 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
+<cmath>k^2_1 + k^2_2 + ... + k^2_n = 2002?</cmath>
 <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>
-[[2001 AMC 12 Problems/Problem 13|Solution]]
+[[2002 AMC 12P Problems/Problem 13|Solution]]
 == 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>
-\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>
-[[2001 AMC 12 Problems/Problem 14|Solution]]
+[[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</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) }\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>
-[[2001 AMC 12 Problems/Problem 15|Solution]]
+[[2002 AMC 12P Problems/Problem 15|Solution]]
 == 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>
-\text{(A) }8!
+\text{(A) }72^\circ
 \qquad
-\text{(B) }2^8 \cdot 8!
+\text{(B) }75^\circ
 \qquad
-\text{(C) }(8!)^2
+\text{(C) }90^\circ
 \qquad
-\text{(D) }\frac {16!}{2^8}
+\text{(D) }108^\circ
 \qquad
-\text{(E) }16!
+\text{(E) }120^\circ
 </math>
-[[2001 AMC 12 Problems/Problem 16|Solution]]
+[[2002 AMC 12P Problems/Problem 16|Solution]]
 == 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) = \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
 <math>
-\text{(A) }\frac {1}{5}
+\text{(A) }1-\sqrt{2}\sin{x}
 \qquad
-\text{(B) }\frac {1}{4}
+\text{(B) }-1+\sqrt{2}\cos{x}
 \qquad
-\text{(C) }\frac {5}{16}
+\text{(C) }\cos{\frac{x}{2}} - \sin{\frac{x}{2}}
 \qquad
-\text{(D) }\frac {3}{8}
+\text{(D) }\cos{x} - \sin{x}
 \qquad
-\text{(E) }\frac {1}{2}
+\text{(E) }\cos{2x}
 </math>
-[[2001 AMC 12 Problems/Problem 17|Solution]]
+[[2002 AMC 12P Problems/Problem 17|Solution]]
 == Problem 18 ==
-A circle centered at <math>A</math> with a radius of 1 and a circle centered at <math>B</math> 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>
-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>
 <math>
-\text{(A) }\frac {1}{3}
+\text{(A) }14
 \qquad
-\text{(B) }\frac {2}{5}
+\text{(B) }21
 \qquad
-\text{(C) }\frac {5}{12}
+\text{(C) }28
 \qquad
-\text{(D) }\frac {4}{9}
+\text{(D) }35
 \qquad
-\text{(E) }\frac {1}{2}
+\text{(E) }49
 </math>
-[[2001 AMC 12 Problems/Problem 18|Solution]]
+[[2002 AMC 12P Problems/Problem 18|Solution]]
 == 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,</math> and <math>CD=5.</math> Find the area of <math>ABCD.</math>
 <math>
-\text{(A) }-11
+\text{(A) }15
 \qquad
-\text{(B) }-10
+\text{(B) }9 \sqrt{3}
 \qquad
-\text{(C) }-9
+\text{(C) }\frac{45 \sqrt{3}}{4}
 \qquad
-\text{(D) }1
+\text{(D) }\frac{47 \sqrt{3}}{4}
 \qquad
-\text{(E) }5
+\text{(E) }15 \sqrt{3}
 </math>
-[[2001 AMC 12 Problems/Problem 19|Solution]]
+[[2002 AMC 12P Problems/Problem 19|Solution]]
 == 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>?
+Let <math>f</math> be a real-valued function such that
+<cmath>f(x) + 2f(\frac{2002}{x}) = 3x</cmath>
+for all <math>x>0.</math> Find <math>f(2).</math>
 <math>
-\text{(A) }7
+\text{(A) }1000
 \qquad
-\text{(B) }9
+\text{(B) }2000
 \qquad
-\text{(C) }10
+\text{(C) }3000
 \qquad
-\text{(D) }12
+\text{(D) }4000
 \qquad
-\text{(E) }16
+\text{(E) }6000
 </math>
-[[2001 AMC 12 Problems/Problem 20|Solution]]
+[[2002 AMC 12P Problems/Problem 20|Solution]]
 == 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:
+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
-<cmath>
+<cmath>2(\log_a{c} + \log_b{c}) = 9\log_{ab}{c}.</cmath>
-\begin{align*}
-ab + a + b & = 524
-\\
-bc + b + c & = 146
-\\
-cd + c + d & = 104
-\end{align*}
-</cmath>
-What is <math>a-d</math>?
+Find the largest possible value of <math>\log_a b.</math>
 <math>
-\text{(A) }4
+\text{(A) }\sqrt{2}
 \qquad
-\text{(B) }6
+\text{(B) }\sqrt{3}
 \qquad
-\text{(C) }8
+\text{(C) }2
 \qquad
-\text{(D) }10
+\text{(D) }\sqrt{6}
 \qquad
-\text{(E) }12
+\text{(E) }3
 </math>
-[[2001 AMC 12 Problems/Problem 21|Solution]]
+[[2002 AMC 12P Problems/Problem 21|Solution]]
 == 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>.
+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 {5}{2}
+\text{(A) }175
 \qquad
-\text{(B) }\frac {35}{12}
+\text{(B) }179.5
 \qquad
-\text{(C) }3
+\text{(C) }182
 \qquad
-\text{(D) }\frac {7}{2}
+\text{(D) }188.5
 \qquad
-\text{(E) }\frac {35}{8}
+\text{(E) }201
 </math>
-[[2001 AMC 12 Problems/Problem 22|Solution]]
+[[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?
+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>
 <math>
-\text{(A) }\frac {1 + i \sqrt {11}}{2}
+\text{(A) }\sqrt{118}
 \qquad
-\text{(B) }\frac {1 + i}{2}
+\text{(B) }\sqrt{210}
 \qquad
-\text{(C) }\frac {1}{2} + i
+\text{(C) }2 \sqrt{210}
 \qquad
-\text{(D) }1 + \frac {i}{2}
+\text{(D) }\sqrt{2002}
 \qquad
-\text{(E) }\frac {1 + i \sqrt {13}}{2}
+\text{(E) }100 \sqrt{2}
 </math>
-[[2001 AMC 12 Problems/Problem 23|Solution]]
+[[2002 AMC 12P Problems/Problem 23|Solution]]
 == 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>.
+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
 <math>
-\text{(A) }54^\circ
+\text{(A) }\sqrt{2}
 \qquad
-\text{(B) }60^\circ
+\text{(B) }\frac{2 \sqrt{2}}{3}
 \qquad
-\text{(C) }72^\circ
+\text{(C) }\frac{\sqrt{6}}{2}
 \qquad
-\text{(D) }75^\circ
+\text{(D) }2
 \qquad
-\text{(E) }90^\circ
+\text{(E) }3
 </math>
-[[2001 AMC 12 Problems/Problem 24|Solution]]
+[[2002 AMC 12P Problems/Problem 24|Solution]]
 == 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?
+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>
 <math>
-\text{(A) }1
+\text{(A) }\frac{1}{2}
 \qquad
-\text{(B) }2
+\text{(B) }\frac{\sqrt{2}}{2}
 \qquad
-\text{(C) }3
+\text{(C) }\frac{\sqrt{3}}{2}
 \qquad
-\text{(D) }4
+\text{(D) }\frac{\sqrt{6}}{2}
 \qquad
-\text{(E) more than }4
+\text{(E) }1
 </math>
-[[2001 AMC 12 Problems/Problem 25|Solution]]
+[[2002 AMC 12P Problems/Problem 25|Solution]]
 == See also ==
-{{AMC12 box|year=2001|before=[[2000 AMC 12 Problems]]|after=[[2002 AMC 12A Problems]]}}
+{{AMC12 box|year=2002|ab=P|before=[[2001 AMC 12 Problems]]|after=[[2002 AMC 12A Problems]]}}
 * [[AMC 12]]
 * [[AMC 12 Problems and Solutions]]
-* [[2001 AMC 12]]
+* [[2002 AMC 12P]]
 * [[Mathematics competition resources]]
 {{MAA Notice}}