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Difference between revisions of "2002 AMC 12P Problems"

(→‎Problem 17)
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(35 intermediate revisions by the same user not shown)
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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>
 +
 
 +
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>
 
<math>
Line 56: 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 65: Line 86:
 
For how many positive integers <math>m</math> is  
 
For how many positive integers <math>m</math> is  
 
<cmath>\frac{2002}{m^2 -2}</cmath>
 
<cmath>\frac{2002}{m^2 -2}</cmath>
 +
 +
a positive integer?
  
 
<math>
 
<math>
Line 81: Line 104:
  
 
== 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 137: Line 160:
 
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 144: Line 177:
  
 
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
 
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>
Line 163: Line 198:
 
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 219: 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>
Line 273: Line 308:
  
 
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
 
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>
 
<math>
 
\text{(A) }1-\sqrt{2}\sin{x}
 
\text{(A) }1-\sqrt{2}\sin{x}
Line 307: Line 343:
 
== Problem 19 ==
 
== Problem 19 ==
  
−
In quadrilateral <math>ABCD</math>, <math>m\angle B = m \angle C = 120^{\circ}, AB=3, BC=4, and 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>
Line 325: Line 361:
 
== Problem 20 ==
 
== 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>
 
<math>
−
\text{(A) }7
+
\text{(A) }1000
 
\qquad
 
\qquad
−
\text{(B) }9
+
\text{(B) }2000
 
\qquad
 
\qquad
−
\text{(C) }10
+
\text{(C) }3000
 
\qquad
 
\qquad
−
\text{(D) }12
+
\text{(D) }4000
 
\qquad
 
\qquad
−
\text{(E) }16
+
\text{(E) }6000
 
</math>
 
</math>
  
Line 343: Line 383:
 
== Problem 21 ==
 
== 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>
 
<math>
−
\text{(A) }4
+
\text{(A) }\sqrt{2}
 
\qquad
 
\qquad
−
\text{(B) }6
+
\text{(B) }\sqrt{3}
 
\qquad
 
\qquad
−
\text{(C) }8
+
\text{(C) }2
 
\qquad
 
\qquad
−
\text{(D) }10
+
\text{(D) }\sqrt{6}
 
\qquad
 
\qquad
−
\text{(E) }12
+
\text{(E) }3
 
</math>
 
</math>
  
−
[[2001 AMC 12 Problems/Problem 21|Solution]]
+
[[2002 AMC 12P Problems/Problem 21|Solution]]
  
 
== Problem 22 ==
 
== 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, 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) }\frac {5}{2}
+
\text{(A) }175
 
\qquad
 
\qquad
−
\text{(B) }\frac {35}{12}
+
\text{(B) }179.5
 
\qquad
 
\qquad
−
\text{(C) }3
+
\text{(C) }182
 
\qquad
 
\qquad
−
\text{(D) }\frac {7}{2}
+
\text{(D) }188.5
 
\qquad
 
\qquad
−
\text{(E) }\frac {35}{8}
+
\text{(E) }201
 
</math>
 
</math>
  
Line 391: Line 423:
 
== Problem 23 ==
 
== 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>
 
<math>
−
\text{(A) }\frac {1 + i \sqrt {11}}{2}
+
\text{(A) }\sqrt{118}
 
\qquad
 
\qquad
−
\text{(B) }\frac {1 + i}{2}
+
\text{(B) }\sqrt{210}
 
\qquad
 
\qquad
−
\text{(C) }\frac {1}{2} + i
+
\text{(C) }2 \sqrt{210}
 
\qquad
 
\qquad
−
\text{(D) }1 + \frac {i}{2}
+
\text{(D) }\sqrt{2002}
 
\qquad
 
\qquad
−
\text{(E) }\frac {1 + i \sqrt {13}}{2}
+
\text{(E) }100 \sqrt{2}
 
</math>
 
</math>
  
Line 409: Line 441:
 
== Problem 24 ==
 
== 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>
 
<math>
−
\text{(A) }54^\circ
+
\text{(A) }\sqrt{2}
 
\qquad
 
\qquad
−
\text{(B) }60^\circ
+
\text{(B) }\frac{2 \sqrt{2}}{3}
 
\qquad
 
\qquad
−
\text{(C) }72^\circ
+
\text{(C) }\frac{\sqrt{6}}{2}
 
\qquad
 
\qquad
−
\text{(D) }75^\circ
+
\text{(D) }2
 
\qquad
 
\qquad
−
\text{(E) }90^\circ
+
\text{(E) }3
 
</math>
 
</math>
  
Line 427: Line 459:
 
== Problem 25 ==
 
== 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>
 
<math>
−
\text{(A) }1
+
\text{(A) }\frac{1}{2}
 
\qquad
 
\qquad
−
\text{(B) }2
+
\text{(B) }\frac{\sqrt{2}}{2}
 
\qquad
 
\qquad
−
\text{(C) }3
+
\text{(C) }\frac{\sqrt{3}}{2}
 
\qquad
 
\qquad
−
\text{(D) }4
+
\text{(D) }\frac{\sqrt{6}}{2}
 
\qquad
 
\qquad
−
\text{(E) more than }4
+
\text{(E) }1
 
</math>
 
</math>
  

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)
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2000 AMC 12 Problems 		Followed by
2002 AMC 12A Problems
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All AMC 12 Problems and Solutions

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