Difference between revisions of "2002 AMC 12P Problems"

Latest revision as of 04:03, 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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Difference between revisions of "2002 AMC 12P Problems"

Latest revision as of 04:03, 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=2002|ab=P}}
 == Problem 1 ==
 Which of the following numbers is a perfect square?
-<math>\text{(A)}\ 4^5 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>
+<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 2|Solution]]
+[[2002 AMC 12P Problems/Problem 1|Solution]]
 == Problem 2 ==
 The function <math>f</math> is given by the table
-If <math>u_0=4</math> and <math>u_{n+1} = f(u_n)</math> for <math>n>=0</math>, find <math>u_2002</math>
+<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>
-<math>\text{(A)}\ 1\qquad \text{(B)}\ 2\qquad \text{(C)}\ 3\qquad \text{(D)}\ 4\qquad \text{(E)}\ 5</math>
+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 dimensions of a rectangular box in inches are all positive integers and the volume of the box is <math>2002</math> in^3. Find the minimum possible sum of the three dimensions.
+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.
 <math>
@@ Line 34: / Line 67: @@
 == Problem 4 ==
 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>
 Find <math>\frac{a}{b}</math>
-<math>\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) }0.4
+\qquad
+\text{(B) }0.5
+\qquad
+\text{(C) }0.6
+\qquad
+\text{(D) }0.7
+\qquad
+\text{(E) }0.8
+</math>
 [[2002 AMC 12P Problems/Problem 4|Solution]]
 == Problem 5 ==
 For how many positive integers <math>m</math> is
 <cmath>\frac{2002}{m^2 -2}</cmath>
-<math>\text{(A)}\ one\qquad \text{(B)}\ two\qquad
+a positive integer?
-\text{(C)}\ three\qquad \text{(D)}\ four\qquad
-\text{(E)}\ five</math>
+<math>
+\text{(A) one}
+\qquad
+\text{(B) two}
+\qquad
+\text{(C) three}
+\qquad
+\text{(D) four}
+\qquad
+\text{(E) more than four}
+</math>
 [[2002 AMC 12P Problems/Problem 5|Solution]]
 == Problem 6 ==
-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?
+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?
 <math>
@@ Line 66: / Line 123: @@
 \text{(E) }\frac{1}{2}
 </math>
 [[2002 AMC 12P Problems/Problem 6|Solution]]
@@ Line 72: / Line 128: @@
 == Problem 7 ==
-How many three-digit numbers have at least one 2 and at least one 3?
+How many three-digit numbers have at least one <math>2</math> and at least one <math>3</math>?
 <math>
@@ Line 108: / Line 164: @@
 == Problem 9 ==
-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?
+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)}\ \sqrt{13} \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>
 [[2002 AMC 12P Problems/Problem 9|Solution]]
@@ Line 116: / Line 182: @@
 == 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) = \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>
@@ Line 127: / Line 195: @@
 \text{(D) }8
 \qquad
-\text{(E) }more than 8
+\text{(E) more than }8
 </math>
@@ Line 136: / Line 204: @@
 Let <math>t_n = \frac{n(n+1)}{2}</math> be the <math>n</math>th triangular number. Find
-<cmath>\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + ... + \frac{1}{t_2002}</cmath>
+<cmath>\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + ... + \frac{1}{t_{2002}}</cmath>
 <math>
@@ Line 157: / Line 225: @@
 <math>
-\text{(A) }one
+\text{(A) one}
 \qquad
-\text{(B) }two
+\text{(B) two}
 \qquad
-\text{(C) }three
+\text{(C) three}
 \qquad
-\text{(D) }four
+\text{(D) four}
 \qquad
-\text{(E) }more than four
+\text{(E) more than four}
 </math>
@@ Line 174: / Line 242: @@
 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
-<math>k^2_1 + k^2_2 + ... + k^2_n = 2002.</math>
+<cmath>k^2_1 + k^2_2 + ... + k^2_n = 2002?</cmath>
 <math>
@@ Line 192: / Line 260: @@
 == Problem 14 ==
-Find <math>i + 2i^2 +3i^3 + ... + 2002i^2002.</math>
+Find <math>i + 2i^2 +3i^3 + . . . + 2002i^{2002}.</math>
 <math>
@@ Line 209: / Line 277: @@
 == Problem 15 ==
-There are <math>1001 red marbles and </math>1001 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>
+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>
@@ Line 230: / Line 299: @@
 <math>
-\text{(A) }72^o
+\text{(A) }72^\circ
 \qquad
-\text{(B) }75^o
+\text{(B) }75^\circ
 \qquad
-\text{(C) }90^o
+\text{(C) }90^\circ
 \qquad
-\text{(D) }108^o
+\text{(D) }108^\circ
 \qquad
-\text{(E) }120^o
+\text{(E) }120^\circ
 </math>
@@ Line 245: / Line 314: @@
 == Problem 17 ==
-Let <math>f(x) =
+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}
+<math>
+\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>
+</math>
 [[2002 AMC 12P Problems/Problem 17|Solution]]
@@ Line 262: / Line 332: @@
 == 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>
-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>
@@ Line 298: / Line 350: @@
 == 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>
@@ Line 316: / Line 368: @@
 == 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>
@@ Line 334: / Line 390: @@
 == 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>
@@ Line 382: / Line 430: @@
 == 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>
@@ Line 400: / Line 448: @@
 == 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>
@@ Line 418: / Line 466: @@
 == 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>
@@ Line 436: / Line 484: @@
 == 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]]
-* [[2002 AMC 12]]
+* [[2002 AMC 12P]]
 * [[Mathematics competition resources]]
 {{MAA Notice}}