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Difference between revisions of "2002 AMC 12P Problems"
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\text{(E) }12
 
\text{(E) }12
 
</math>
 
</math>
 +
 
[[2002 AMC 12P Problems/Problem 8|Solution]]
 
[[2002 AMC 12P Problems/Problem 8|Solution]]
  
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== 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>]<math> is it true that
+
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
  
−
</math>
+
<math>
 
\text{(A) }2
 
\text{(A) }2
 
\qquad
 
\qquad
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\qquad
 
\qquad
 
\text{(E) }more than 8
 
\text{(E) }more than 8
−
<math>
+
</math>
  
 
[[2002 AMC 12P Problems/Problem 10|Solution]]
 
[[2002 AMC 12P Problems/Problem 10|Solution]]
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== Problem 11 ==
 
== Problem 11 ==
  
−
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>
 
\text{(A) }\frac {4003}{2003}
 
\text{(A) }\frac {4003}{2003}
 
\qquad
 
\qquad
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\qquad
 
\qquad
 
\text{(E) }2
 
\text{(E) }2
−
<math>
+
</math>
  
 
[[2002 AMC 12P Problems/Problem 11|Solution]]
 
[[2002 AMC 12P Problems/Problem 11|Solution]]
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== Problem 12 ==
 
== Problem 12 ==
  
−
For how many positive integers </math>n<math> is </math>n^3 - 8n^2 + 20n - 13<math> a prime number?
+
For how many positive integers <math>n</math> is <math>n^3 - 8n^2 + 20n - 13</math> a prime number?
  
−
</math>
+
<math>
 
\text{(A) }one
 
\text{(A) }one
 
\qquad
 
\qquad
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\qquad
 
\qquad
 
\text{(E) }more than four
 
\text{(E) }more than four
−
<math>
+
</math>
  
 
[[2002 AMC 12P Problems/Problem 12|Solution]]
 
[[2002 AMC 12P Problems/Problem 12|Solution]]
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== Problem 13 ==
 
== Problem 13 ==
  
−
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
+
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>
+
<math>k^2_1 + k^2_2 + ... + k^2_n = 2002.</math>
  
−
</math>
+
<math>
 
\text{(A) }14
 
\text{(A) }14
 
\qquad
 
\qquad
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\qquad
 
\qquad
 
\text{(E) }18
 
\text{(E) }18
−
<math>
+
</math>
  
 
[[2002 AMC 12P Problems/Problem 13|Solution]]
 
[[2002 AMC 12P Problems/Problem 13|Solution]]
Line 191: Line 192:
 
== 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>
 
\text{(A) }-999 + 1002i
 
\text{(A) }-999 + 1002i
 
\qquad
 
\qquad
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\qquad
 
\qquad
 
\text{(E) }i
 
\text{(E) }i
−
<math>
+
</math>
  
 
[[2002 AMC 12P Problems/Problem 14|Solution]]
 
[[2002 AMC 12P Problems/Problem 14|Solution]]
  
 
== 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 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>
  
−
</math>
+
<math>
 
\text{(A) }0
 
\text{(A) }0
 
\qquad
 
\qquad
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\qquad
 
\qquad
 
\text{(E) }\frac{1}{1000}
 
\text{(E) }\frac{1}{1000}
−
<math>
+
</math>
  
 
[[2002 AMC 12P Problems/Problem 15|Solution]]
 
[[2002 AMC 12P Problems/Problem 15|Solution]]
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== Problem 16 ==
 
== Problem 16 ==
  
−
The altitudes of a triangle are </math>12, 15,<math> and </math>20.<math> The largest angle in this triangle is
+
The altitudes of a triangle are <math>12, 15,</math> and <math>20.</math> The largest angle in this triangle is
  
−
</math>
+
<math>
 
\text{(A) }72^o
 
\text{(A) }72^o
 
\qquad
 
\qquad
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\qquad
 
\qquad
 
\text{(E) }120^o
 
\text{(E) }120^o
−
<math>
+
</math>
  
 
[[2002 AMC 12P Problems/Problem 16|Solution]]
 
[[2002 AMC 12P Problems/Problem 16|Solution]]
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== Problem 17 ==
 
== Problem 17 ==
  
−
Let </math>f(x) =  
+
Let <math>f(x) =  
−
<math>
+
</math>
 
\text{(A) }\frac {1}{5}
 
\text{(A) }\frac {1}{5}
 
\qquad
 
\qquad
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\qquad
 
\qquad
 
\text{(E) }\frac {1}{2}
 
\text{(E) }\frac {1}{2}
−
</math>
+
<math>
  
 
[[2002 AMC 12P Problems/Problem 17|Solution]]
 
[[2002 AMC 12P Problems/Problem 17|Solution]]
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== Problem 18 ==
 
== 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
+
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
  
 
<asy>
 
<asy>

Revision as of 20:51, 29 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?

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

Solution

Problem 2

The function $f$ is given by the table

If $u_0=4$ and $u_{n+1} = f(u_n)$ for $n>=0$, find $u_2002$

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

Solution

Problem 3

The dimensions of a rectangular box in inches are all positive integers and the volume of the box is $2002$ in^3. Find the minimum possible sum of the three dimensions.

$\text{(A) }36 \qquad \text{(B) }38 \qquad \text{(C) }42 \qquad \text{(D) }44 \qquad \text{(E) }92$

Solution

Problem 4

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

$\text{(A)}\ 0.4\qquad \text{(B)}\ 0.5\qquad \text{(C)}\ 0.6\qquad \text{(D)}\ 0.7\qquad \text{(E)}\ 0.8$

Solution

Problem 5

For how many positive integers $m$ is \[\frac{2002}{m^2 -2}\]

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

Solution

Problem 6

Participation in the local soccer league this year is $10%$ (Error compiling LaTeX. Unknown error_msg) higher than last year. The number of males increased by $5%$ (Error compiling LaTeX. Unknown error_msg) and the number of females increased by $20%$ (Error compiling LaTeX. Unknown error_msg). What fraction of the soccer league is now female?

$\text{(A) }\frac{1}{3} \qquad \text{(B) }\frac{4}{11} \qquad \text{(C) }\frac{2}{5} \qquad \text{(D) }\frac{4}{9} \qquad \text{(E) }\frac{1}{2}$


Solution

Problem 7

How many three-digit numbers have at least one 2 and at least one 3?

$\text{(A) }52 \qquad \text{(B) }54 \qquad \text{(C) }56 \qquad \text{(D) }58 \qquad \text{(E) }60$

Solution

Problem 8

Let $AB$ be a segment of length $26$, and let points $C$ and $D$ be located on $AB$ such that $AC=1$ and $AD=8$. Let $E$ and $F$ be points on one of the semicircles with diameter $AB$ for which $EC$ and $FD$ are perpendicular to $AB$. Find $EF.$

$\text{(A) }5 \qquad \text{(B) }5 \sqrt{2} \qquad \text{(C) }7 \qquad \text{(D) }7 \sqrt{2} \qquad \text{(E) }12$

Solution

Problem 9

Two walls and the ceiling of a room meet at right angles at point $P.$ A fly is in the air one meter from one wall, eight meters from the other wall, and nine meters from point $P$. How many meters is the fly from the ceiling?

$\text{(A)}\ \sqrt{13} \qquad \text{(B)}\ 2 \qquad \text{(C)}\ \frac52 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ \frac{18}5$

Solution

Problem 10

Let $f_n (x) = sin^n x + cos^n x.$ For how many $x$ in [$0,π$ (Error compiling LaTeX. Unknown error_msg)] is it true that

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

Solution

Problem 11

Let $t_n = \frac{n(n+1)}{2}$ be the $n$th triangular number. Find

\[\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + ... + \frac{1}{t_2002}\]

$\text{(A) }\frac {4003}{2003} \qquad \text{(B) }\frac {2001}{1001} \qquad \text{(C) }\frac {4004}{2003} \qquad \text{(D) }\frac {4001}{2001} \qquad \text{(E) }2$

Solution

Problem 12

For how many positive integers $n$ is $n^3 - 8n^2 + 20n - 13$ a prime number?

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

Solution

Problem 13

What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1, k_2, ... k_n$ for which

$k^2_1 + k^2_2 + ... + k^2_n = 2002.$

$\text{(A) }14 \qquad \text{(B) }15 \qquad \text{(C) }16 \qquad \text{(D) }17 \qquad \text{(E) }18$

Solution

Problem 14

Find $i + 2i^2 +3i^3 + ... + 2002i^2002.$

$\text{(A) }-999 + 1002i \qquad \text{(B) }-1002 + 999i \qquad \text{(C) }-1001 + 1000i \qquad \text{(D) }-1002 + 1001i \qquad \text{(E) }i$

Solution

Problem 15

There are $1001 red marbles and$1001 black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random from the box are the same color, and let $P_d$ be the probability that they are different colors. Find $|P_s-P_d|.$

$\text{(A) }0 \qquad \text{(B) }\frac{1}{2002} \qquad \text{(C) }\frac{1}{2001} \qquad \text{(D) }\frac {2}{2001} \qquad \text{(E) }\frac{1}{1000}$

Solution

Problem 16

The altitudes of a triangle are $12, 15,$ and $20.$ The largest angle in this triangle is

$\text{(A) }72^o \qquad \text{(B) }75^o \qquad \text{(C) }90^o \qquad \text{(D) }108^o \qquad \text{(E) }120^o$

Solution

Problem 17

Let $f(x) =$ \text{(A) }\frac {1}{5} \qquad \text{(B) }\frac {1}{4} \qquad \text{(C) }\frac {5}{16} \qquad \text{(D) }\frac {3}{8} \qquad \text{(E) }\frac {1}{2} $[[2002 AMC 12P Problems/Problem 17|Solution]]

== Problem 18 ==

A circle centered at$ (Error compiling LaTeX. Unknown error_msg)A$with a radius of 1 and a circle centered at$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

[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]

$\text{(A) }\frac {1}{3} \qquad \text{(B) }\frac {2}{5} \qquad \text{(C) }\frac {5}{12} \qquad \text{(D) }\frac {4}{9} \qquad \text{(E) }\frac {1}{2}$

Solution

Problem 19

The polynomial $P(x)=x^3+ax^2+bx+c$ 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 $y$-intercept of the graph of $y=P(x)$ is 2, what is $b$?

$\text{(A) }-11 \qquad \text{(B) }-10 \qquad \text{(C) }-9 \qquad \text{(D) }1 \qquad \text{(E) }5$

Solution

Problem 20

Points $A = (3,9)$, $B = (1,1)$, $C = (5,3)$, and $D=(a,b)$ lie in the first quadrant and are the vertices of quadrilateral $ABCD$. The quadrilateral formed by joining the midpoints of $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$ is a square. What is the sum of the coordinates of point $D$?

$\text{(A) }7 \qquad \text{(B) }9 \qquad \text{(C) }10 \qquad \text{(D) }12 \qquad \text{(E) }16$

Solution

Problem 21

Four positive integers $a$, $b$, $c$, and $d$ have a product of $8!$ and satisfy:

\begin{align*} ab + a + b & = 524 \\ bc + b + c & = 146 \\ cd + c + d & = 104 \end{align*}

What is $a-d$?

$\text{(A) }4 \qquad \text{(B) }6 \qquad \text{(C) }8 \qquad \text{(D) }10 \qquad \text{(E) }12$

Solution

Problem 22

In rectangle $ABCD$, points $F$ and $G$ lie on $AB$ so that $AF=FG=GB$ and $E$ is the midpoint of $\overline{DC}$. Also, $\overline{AC}$ intersects $\overline{EF}$ at $H$ and $\overline{EG}$ at $J$. The area of the rectangle $ABCD$ is $70$. Find the area of triangle $EHJ$.

$\text{(A) }\frac {5}{2} \qquad \text{(B) }\frac {35}{12} \qquad \text{(C) }3 \qquad \text{(D) }\frac {7}{2} \qquad \text{(E) }\frac {35}{8}$

Solution

Problem 23

A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros, both of which are integers. Which of the following can also be a zero of the polynomial?

$\text{(A) }\frac {1 + i \sqrt {11}}{2} \qquad \text{(B) }\frac {1 + i}{2} \qquad \text{(C) }\frac {1}{2} + i \qquad \text{(D) }1 + \frac {i}{2} \qquad \text{(E) }\frac {1 + i \sqrt {13}}{2}$

Solution

Problem 24

In $\triangle ABC$, $\angle ABC=45^\circ$. Point $D$ is on $\overline{BC}$ so that $2\cdot BD=CD$ and $\angle DAB=15^\circ$. Find $\angle ACB$.

$\text{(A) }54^\circ \qquad \text{(B) }60^\circ \qquad \text{(C) }72^\circ \qquad \text{(D) }75^\circ \qquad \text{(E) }90^\circ$

Solution

Problem 25

Consider sequences of positive real numbers of the form $x, 2000, y, \dots$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $x$ does the term 2001 appear somewhere in the sequence?

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

Solution

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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