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Difference between revisions of "2002 AMC 12P Problems"
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Which of the following numbers is a perfect square?
 
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 ==
 
== Problem 2 ==
 
The function <math>f</math> is given by the table
 
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]]
 
[[2002 AMC 12P Problems/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== 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>
 
<math>
Line 36: Line 66:
 
Let <math>a</math> and <math>b</math> be distinct real numbers for which
 
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>
 
<cmath>\frac{a}{b} + \frac{a+10b}{b+10a} = 2.</cmath>
 +
 
Find <math>\frac{a}{b}</math>
 
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]]
 
[[2002 AMC 12P Problems/Problem 4|Solution]]
Line 46: Line 87:
 
<cmath>\frac{2002}{m^2 -2}</cmath>
 
<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]]
 
[[2002 AMC 12P Problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== 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>
 
<math>
Line 66: Line 117:
 
\text{(E) }\frac{1}{2}
 
\text{(E) }\frac{1}{2}
 
</math>
 
</math>
−
 
  
 
[[2002 AMC 12P Problems/Problem 6|Solution]]
 
[[2002 AMC 12P Problems/Problem 6|Solution]]
Line 72: Line 122:
 
== Problem 7 ==
 
== 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>
 
<math>
Line 108: Line 158:
 
== Problem 9 ==
 
== 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]]
 
[[2002 AMC 12P Problems/Problem 9|Solution]]
Line 116: Line 176:
 
== Problem 10 ==
 
== 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>
 
<math>
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\text{(D) }8
 
\text{(D) }8
 
\qquad
 
\qquad
−
\text{(E) }more than 8
+
\text{(E) more than }8
 
</math>
 
</math>
  
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Let <math>t_n = \frac{n(n+1)}{2}</math> be the <math>n</math>th triangular number. Find
 
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>
 
<math>
Line 157: Line 219:
  
 
<math>
 
<math>
−
\text{(A) }one
+
\text{(A) one}
 
\qquad
 
\qquad
−
\text{(B) }two
+
\text{(B) two}
 
\qquad
 
\qquad
−
\text{(C) }three
+
\text{(C) three}
 
\qquad
 
\qquad
−
\text{(D) }four
+
\text{(D) four}
 
\qquad
 
\qquad
−
\text{(E) }more than four
+
\text{(E) more than four}
 
</math>
 
</math>
  
Line 192: Line 254:
 
== Problem 14 ==
 
== Problem 14 ==
  
−
Find <math>i + 2i^2 +3i^3 + ... + 2002i^{2002}.</math>
+
Find <math>i + 2i^2 +3i^3 + . . . + 2002i^{2002}.</math>
  
 
<math>
 
<math>
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== Problem 15 ==
 
== 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>
 
<math>
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<math>
 
<math>
−
\text{(A) }72^{\circ}
+
\text{(A) }72^\circ
 
\qquad
 
\qquad
−
\text{(B) }75^{\circ}
+
\text{(B) }75^\circ
 
\qquad
 
\qquad
−
\text{(C) }90^{\circ}
+
\text{(C) }90^\circ
 
\qquad
 
\qquad
−
\text{(D) }108^{\circ}
+
\text{(D) }108^\circ
 
\qquad
 
\qquad
−
\text{(E) }120^{\circ}
+
\text{(E) }120^\circ
 
</math>
 
</math>
  
Line 245: Line 307:
 
== Problem 17 ==
 
== 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
 
\qquad
−
\text{(B) }\frac {1}{4}
+
\text{(B) }-1+\sqrt{2}\cos{x}
 
\qquad
 
\qquad
−
\text{(C) }\frac {5}{16}
+
\text{(C) }\cos{\frac{x}{2}} - \sin{\frac{x}{2}}
 
\qquad
 
\qquad
−
\text{(D) }\frac {3}{8}
+
\text{(D) }\cos{x} - \sin{x}
 
\qquad
 
\qquad
−
\text{(E) }\frac {1}{2}
+
\text{(E) }\cos{2x}
−
<math>
+
</math>
  
 
[[2002 AMC 12P Problems/Problem 17|Solution]]
 
[[2002 AMC 12P Problems/Problem 17|Solution]]
Line 262: Line 325:
 
== Problem 18 ==
 
== Problem 18 ==
  
−
If </math>a,b,c<math> are real numbers such that </math>a^2 + 2b=7, 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
 
\text{(A) }14
 
\qquad
 
\qquad
Line 274: Line 337:
 
\qquad
 
\qquad
 
\text{(E) }49
 
\text{(E) }49
−
<math>
+
</math>
  
 
[[2002 AMC 12P Problems/Problem 18|Solution]]
 
[[2002 AMC 12P Problems/Problem 18|Solution]]
Line 280: Line 343:
 
== Problem 19 ==
 
== 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>
+
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>
+
 
 +
<math>
 
\text{(A) }15
 
\text{(A) }15
 
\qquad
 
\qquad
Line 291: Line 355:
 
\qquad
 
\qquad
 
\text{(E) }15 \sqrt{3}
 
\text{(E) }15 \sqrt{3}
−
<math>
+
</math>
  
 
[[2002 AMC 12P Problems/Problem 19|Solution]]
 
[[2002 AMC 12P Problems/Problem 19|Solution]]
Line 297: Line 361:
 
== Problem 20 ==
 
== Problem 20 ==
  
−
Let </math>f<math> be a real-valued function such that
+
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>
+
<cmath>f(x) + 2f(\frac{2002}{x}) = 3x</cmath>
  
−
</math>
+
for all <math>x>0.</math> Find <math>f(2).</math>
 +
 
 +
<math>
 
\text{(A) }1000
 
\text{(A) }1000
 
\qquad
 
\qquad
Line 311: Line 377:
 
\qquad
 
\qquad
 
\text{(E) }6000
 
\text{(E) }6000
−
<math>
+
</math>
  
 
[[2002 AMC 12P Problems/Problem 20|Solution]]
 
[[2002 AMC 12P Problems/Problem 20|Solution]]
Line 317: Line 383:
 
== Problem 21 ==
 
== 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>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>2(log_a c + log_b c)=9log_{ab} c</cmath>
 
  
−
</math>
+
<cmath>2(\log_a{c} + \log_b{c}) = 9\log_{ab}{c}.</cmath>
 +
 
 +
Find the largest possible value of <math>\log_a b.</math>
 +
 
 +
<math>
 
\text{(A) }\sqrt{2}
 
\text{(A) }\sqrt{2}
 
\qquad
 
\qquad
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\text{(D) }\sqrt{6}
 
\text{(D) }\sqrt{6}
 
\qquad
 
\qquad
−
\text{(E) }\sqrt{3}
+
\text{(E) }3
−
<math>
+
</math>
  
−
[[2001 AMC 12 Problems/Problem 21|Solution]]
+
[[2002 AMC 12P Problems/Problem 21|Solution]]
  
 
== Problem 22 ==
 
== Problem 22 ==
  
−
Under the new AMC </math>10<math>, </math>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 scores that can be obtained in exactly three ways. What is their sum?
+
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, 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 answers, and also with <math>12</math> correct answers and <math>13</math> unanswered questions. There are scores that can be obtained in exactly three ways. What is their sum?
  
−
</math>
+
<math>
 
\text{(A) }175
 
\text{(A) }175
 
\qquad
 
\qquad
Line 348: Line 417:
 
\qquad
 
\qquad
 
\text{(E) }201
 
\text{(E) }201
−
<math>
+
</math>
  
 
[[2002 AMC 12P Problems/Problem 22|Solution]]
 
[[2002 AMC 12P Problems/Problem 22|Solution]]
Line 354: Line 423:
 
== Problem 23 ==
 
== 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>
+
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>
+
<math>
 
\text{(A) }\sqrt{118}
 
\text{(A) }\sqrt{118}
 
\qquad
 
\qquad
Line 366: Line 435:
 
\qquad
 
\qquad
 
\text{(E) }100 \sqrt{2}
 
\text{(E) }100 \sqrt{2}
−
<math>
+
</math>
  
 
[[2002 AMC 12P Problems/Problem 23|Solution]]
 
[[2002 AMC 12P Problems/Problem 23|Solution]]
Line 372: Line 441:
 
== Problem 24 ==
 
== 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
+
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>
+
<math>
−
\text{(A) }\sqrt {2}
+
\text{(A) }\sqrt{2}
 
\qquad
 
\qquad
 
\text{(B) }\frac{2 \sqrt{2}}{3}
 
\text{(B) }\frac{2 \sqrt{2}}{3}
 
\qquad
 
\qquad
−
\text{(C) }\frac{\sqrt(6)}{2}
+
\text{(C) }\frac{\sqrt{6}}{2}
 
\qquad
 
\qquad
 
\text{(D) }2
 
\text{(D) }2
 
\qquad
 
\qquad
 
\text{(E) }3
 
\text{(E) }3
−
<math>
+
</math>
  
 
[[2002 AMC 12P Problems/Problem 24|Solution]]
 
[[2002 AMC 12P Problems/Problem 24|Solution]]
Line 390: Line 459:
 
== Problem 25 ==
 
== 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  
+
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>
+
<math>
 
\text{(A) }\frac{1}{2}
 
\text{(A) }\frac{1}{2}
 
\qquad
 
\qquad
Line 402: Line 471:
 
\qquad
 
\qquad
 
\text{(E) }1
 
\text{(E) }1
−
$
+
</math>
  
 
[[2002 AMC 12P Problems/Problem 25|Solution]]
 
[[2002 AMC 12P Problems/Problem 25|Solution]]

Latest revision as of 01:46, 31 December 2023

2002 AMC 12P (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. 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.
  2. 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.
  3. 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).
  4. Figures are not necessarily drawn to scale.
  5. 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, a score of $104.5$ can be obtained with $17$ correct answers, $1$ unanswered question, and $7$ incorrect answers, and also with $12$ correct answers and $13$ unanswered questions. There are 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

See also

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

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