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Difference between revisions of "2002 AMC 12P Problems"
Line 174: Line 174:
 
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>
+
<cmath>k^2_1 + k^2_2 + ... + k^2_n = 2002?</cmath>
  
 
<math>
 
<math>
Line 192: 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>
Line 230: Line 230:
  
 
<math>
 
<math>
−
\text{(A) }72^o
+
\text{(A) }72^{\circ}
 
\qquad
 
\qquad
−
\text{(B) }75^o
+
\text{(B) }75^{\circ}
 
\qquad
 
\qquad
−
\text{(C) }90^o
+
\text{(C) }90^{\circ}
 
\qquad
 
\qquad
−
\text{(D) }108^o
+
\text{(D) }108^{\circ}
 
\qquad
 
\qquad
−
\text{(E) }120^o
+
\text{(E) }120^{\circ}
 
</math>
 
</math>
  
Line 262: Line 262:
 
== Problem 18 ==
 
== 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, b^2 + 4c = -7,<math> and </math>c^2+6a = -14,<math> find </math>a^2+b^2+c^2.<math>
  
−
<asy>
+
</math>
−
unitsize(0.75cm);
+
\text{(A) }14
−
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}
 
 
\qquad
 
\qquad
−
\text{(B) }\frac {2}{5}
+
\text{(B) }21
 
\qquad
 
\qquad
−
\text{(C) }\frac {5}{12}
+
\text{(C) }28
 
\qquad
 
\qquad
−
\text{(D) }\frac {4}{9}
+
\text{(D) }35
 
\qquad
 
\qquad
−
\text{(E) }\frac {1}{2}
+
\text{(E) }49
−
</math>
+
<math>
  
 
[[2002 AMC 12P Problems/Problem 18|Solution]]
 
[[2002 AMC 12P Problems/Problem 18|Solution]]
Line 298: Line 280:
 
== Problem 19 ==
 
== 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>
−
<math>
+
\text{(A) }15
−
\text{(A) }-11
 
 
\qquad
 
\qquad
−
\text{(B) }-10
+
\text{(B) }9 \sqrt{3}
 
\qquad
 
\qquad
−
\text{(C) }-9
+
\text{(C) }\frac{45 \sqrt{3}}{4}
 
\qquad
 
\qquad
−
\text{(D) }1
+
\text{(D) }\frac{47 \sqrt{3}}{4}
 
\qquad
 
\qquad
−
\text{(E) }5
+
\text{(E) }15 \sqrt{3}
−
</math>
+
<math>
  
 
[[2002 AMC 12P Problems/Problem 19|Solution]]
 
[[2002 AMC 12P Problems/Problem 19|Solution]]
Line 316: Line 297:
 
== 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>
  
 
[[2002 AMC 12P Problems/Problem 20|Solution]]
 
[[2002 AMC 12P Problems/Problem 20|Solution]]
Line 334: Line 317:
 
== 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>2(log_a c + log_b c)=9log_{ab} c</cmath>
  
−
<cmath>
+
</math>
−
\begin{align*}
+
\text{(A) }\sqrt{2}
−
ab + a + b & = 524
 
−
\
 
−
bc + b + c & = 146
 
−
\
 
−
cd + c + d & = 104
 
−
\end{align*}
 
−
</cmath>
 
−
 
 
−
What is <math>a-d</math>?
 
−
 
 
−
<math>
 
−
\text{(A) }4
 
 
\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) }\sqrt{3}
−
</math>
+
<math>
  
 
[[2001 AMC 12 Problems/Problem 21|Solution]]
 
[[2001 AMC 12 Problems/Problem 21|Solution]]
Line 364: Line 336:
 
== 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<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?
  
−
<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>
  
 
[[2002 AMC 12P Problems/Problem 22|Solution]]
 
[[2002 AMC 12P Problems/Problem 22|Solution]]
Line 382: Line 354:
 
== 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>
  
 
[[2002 AMC 12P Problems/Problem 23|Solution]]
 
[[2002 AMC 12P Problems/Problem 23|Solution]]
Line 400: Line 372:
 
== 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>
  
 
[[2002 AMC 12P Problems/Problem 24|Solution]]
 
[[2002 AMC 12P Problems/Problem 24|Solution]]
Line 418: Line 390:
 
== 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>
+
</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>
+
$
  
 
[[2002 AMC 12P Problems/Problem 25|Solution]]
 
[[2002 AMC 12P Problems/Problem 25|Solution]]

Revision as of 22:09, 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

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

Solution

Problem 17

Let $f(x) =$ \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 ==

If$ (Error compiling LaTeX. Unknown error_msg)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.$$ (Error compiling LaTeX. Unknown error_msg) \text{(A) }14 \qquad \text{(B) }21 \qquad \text{(C) }28 \qquad \text{(D) }35 \qquad \text{(E) }49 $[[2002 AMC 12P Problems/Problem 18|Solution]]

== Problem 19 ==

In quadrilateral$ (Error compiling LaTeX. Unknown error_msg)ABCD$,$m\angle B = m\angle C = 120^{\circ}, AB=3, BC=4,$and$CD=5.$Find the area of$ABCD.$$ (Error compiling LaTeX. Unknown error_msg) \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} $[[2002 AMC 12P Problems/Problem 19|Solution]]

== Problem 20 ==

Let$ (Error compiling LaTeX. Unknown error_msg)f$be a real-valued function such that <cmath>f(x) + 2f(\frac{2002}{x})=3x</cmath> for all$x>0$. Find$f(2).$$ (Error compiling LaTeX. Unknown error_msg) \text{(A) }1000 \qquad \text{(B) }2000 \qquad \text{(C) }3000 \qquad \text{(D) }4000 \qquad \text{(E) }6000 $[[2002 AMC 12P Problems/Problem 20|Solution]]

== Problem 21 ==

Let$ (Error compiling LaTeX. Unknown error_msg)a$and$b$be real numbers greater than$1$for which there exists a positive real number$c$, different from$1$, such that <cmath>2(log_a c + log_b c)=9log_{ab} c</cmath>$ \text{(A) }\sqrt{2} \qquad \text{(B) }\sqrt{3} \qquad \text{(C) }2 \qquad \text{(D) }\sqrt{6} \qquad \text{(E) }\sqrt{3} $[[2001 AMC 12 Problems/Problem 21|Solution]]

== Problem 22 ==

Under the new AMC$ (Error compiling LaTeX. Unknown error_msg)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 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 $[[2002 AMC 12P Problems/Problem 22|Solution]]

== Problem 23 ==

The equation$ (Error compiling LaTeX. Unknown error_msg)z(z+i)(z+3i)=2002i$has a zero of the form$a+bi,$where$a$and$b$are positive real numbers. Find$a.$$ (Error compiling LaTeX. Unknown error_msg) \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} $[[2002 AMC 12P Problems/Problem 23|Solution]]

== Problem 24 ==

Let$ (Error compiling LaTeX. Unknown error_msg)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 $[[2002 AMC 12P Problems/Problem 24|Solution]]

== Problem 25 ==

Let$ (Error compiling LaTeX. Unknown error_msg)a$and$b$be real numbers such that$sin a + sin b = \frac{\sqrt{2}}{2}$and$ \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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