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Difference between revisions of "2006 AMC 12B Problems"
m (→‎Problem 14: typo fix)
 
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 +
{{AMC12 Problems|year=2006|ab=B}}
 
== Problem 1 ==
 
== Problem 1 ==
 
What is <math>( - 1)^1 + ( - 1)^2 + \cdots + ( - 1)^{2006}</math>?
 
What is <math>( - 1)^1 + ( - 1)^2 + \cdots + ( - 1)^{2006}</math>?
Line 25: Line 26:
  
 
== Problem 4 ==
 
== Problem 4 ==
−
Mary is about to pay for five items at the grocery store.  The prices of the items are <math>\</math> <math>7.99</math>, <math>\</math> <math>4.99</math>, <math>\</math> <math>2.99</math>, <math>\</math> <math>1.99</math>, and <math>\</math> <math>0.99</math>.  Mary will pay with a twenty-dollar bill.  Which of the following is closest to the percentage of the <math>\</math> <math>20.00</math> that she will receive in change?
+
Mary is about to pay for five items at the grocery store.  The prices of the items are <math>\textdollar7.99</math>, <math>\textdollar4.99</math>, <math>\textdollar2.99</math>, <math>\textdollar1.99</math>, and <math>\textdollar0.99</math>.  Mary will pay with a twenty-dollar bill.  Which of the following is closest to the percentage of the <math>\textdollar20.00</math> that she will receive in change?
  
 
<math>
 
<math>
Line 43: Line 44:
  
 
== Problem 6 ==
 
== Problem 6 ==
−
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade.  There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar.  Water contains no calories.  How many calories are in 200 grams of her lemonade.
+
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade.  There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar.  Water contains no calories.  How many calories are in 200 grams of her lemonade?
  
 
<math>
 
<math>
Line 79: Line 80:
  
 
== Problem 10 ==
 
== Problem 10 ==
−
Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?
+
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?
  
 
<math>
 
<math>
−
\text {(A) } \frac 67 \qquad \text {(B) } \frac{13}{14} \qquad \text {(C) } 1 \qquad \text {(D) } \frac{14}{13} \qquad \text {(E) } \frac 76
+
\text {(A) } 43 \qquad \text {(B) } 44 \qquad \text {(C) } 45 \qquad \text {(D) } 46 \qquad \text {(E) } 47
 
</math>
 
</math>
  
Line 97: Line 98:
  
 
== Problem 12 ==
 
== Problem 12 ==
−
The parabola <math>y=ax^2+bx+c</math> has vertex <math>(p,p)</math> and <math>y</math>-intercept <math>(0,-p)</math>, where <math>p\new 0</math>. What is <math>b</math>?
+
The parabola <math>y=ax^2+bx+c</math> has vertex <math>(p,p)</math> and <math>y</math>-intercept <math>(0,-p)</math>, where <math>p\ne 0</math>. What is <math>b</math>?
  
 
<math>
 
<math>
Line 106: Line 107:
  
 
== Problem 13 ==
 
== Problem 13 ==
−
{{problem}}
+
Rhombus <math>ABCD</math> is similar to rhombus <math>BFDE</math>. The area of rhombus <math>ABCD</math> is 24, and <math>\angle BAD = 60^\circ</math>. What is the area of rhombus <math>BFDE</math>?
 +
 
 +
<asy> defaultpen(linewidth(0.7)+fontsize(11));
 +
unitsize(2cm);
 +
pair A=origin, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/sqrt(3));
 +
pair point=(3/2, sqrt(3)/2);
 +
draw(B--C--D--A--B--F--D--E--B);
 +
label("$A$", A, dir(point--A));
 +
label("$B$", B, dir(point--B));
 +
label("$C$", C, dir(point--C));
 +
label("$D$", D, dir(point--D));
 +
label("$E$", E, dir(point--E));
 +
label("$F$", F, dir(point--F));
 +
</asy>
 +
 
 +
<math> \textrm{(A) } 6 \qquad \textrm{(B) } 4\sqrt {3} \qquad \textrm{(C) } 8 \qquad \textrm{(D) } 9 \qquad \textrm{(E) } 6\sqrt {3}</math>
  
 
[[2006 AMC 12B Problems/Problem 13|Solution]]
 
[[2006 AMC 12B Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
−
{{problem}}
+
Elmo makes <math>N</math> sandwiches for a fundraiser. For each sandwich he uses <math>B</math> globs of peanut butter at <math>4</math> cents per glob and <math>J</math> blobs of jam at <math>5</math> cents per blob. The cost of the peanut butter and jam to make all the sandwiches is <math>\textdollar 2.53</math>. Assume that <math>B</math>, <math>J</math> and <math>N</math> are all positive integers with <math>N>1</math>. What is the cost of the jam Elmo uses to make the sandwiches?
 +
 
 +
<math>
 +
\mathrm{(A)}\ 1.05
 +
\qquad
 +
\mathrm{(B)}\ 1.25
 +
\qquad
 +
\mathrm{(C)}\ 1.45
 +
\qquad
 +
\mathrm{(D)}\ 1.65
 +
\qquad
 +
\mathrm{(E)}\ 1.85
 +
</math>
  
 
[[2006 AMC 12B Problems/Problem 14|Solution]]
 
[[2006 AMC 12B Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==
−
{{problem}}
+
Circles with centers <math> O</math> and <math> P</math> have radii 2 and 4, respectively, and are externally tangent.  Points <math> A</math> and <math> B</math> are on the circle centered at <math> O</math>, and points <math> C</math> and <math> D</math> are on the circle centered at <math> P</math>, such that <math> \overline{AD}</math> and <math> \overline{BC}</math> are common external tangents to the circles.  What is the area of hexagon <math> AOBCPD</math>?
 +
 
 +
<asy>
 +
unitsize(0.4 cm); defaultpen(linewidth(0.7) + fontsize(11));
 +
pair A, B, C, D;
 +
pair[] O;
 +
O[1] = (6,0);
 +
O[2] = (12,0);
 +
A = (32/6,8*sqrt(2)/6);
 +
B = (32/6,-8*sqrt(2)/6);
 +
C = 2*B;
 +
D = 2*A;
 +
draw(Circle(O[1],2));
 +
draw(Circle(O[2],4));
 +
draw((0.7*A)--(1.2*D));
 +
draw((0.7*B)--(1.2*C));
 +
draw(O[1]--O[2]);
 +
draw(A--O[1]);
 +
draw(B--O[1]);
 +
draw(C--O[2]);
 +
draw(D--O[2]);
 +
label("$A$", A, NW);
 +
label("$B$", B, SW);
 +
label("$C$", C, SW);
 +
label("$D$", D, NW);
 +
dot("$O$", O[1], SE);
 +
dot("$P$", O[2], SE);
 +
label("$2$", (A + O[1])/2, E);
 +
label("$4$", (D + O[2])/2, E);</asy>
 +
 
 +
<math> \textbf{(A) } 18\sqrt {3} \qquad \textbf{(B) } 24\sqrt {2} \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 24\sqrt {3} \qquad \textbf{(E) } 32\sqrt {2}</math>
  
 
[[2006 AMC 12B Problems/Problem 15|Solution]]
 
[[2006 AMC 12B Problems/Problem 15|Solution]]
  
 
== Problem 16 ==
 
== Problem 16 ==
−
{{problem}}
+
 
 +
Regular hexagon <math>ABCDEF</math> has vertices <math>A</math> and <math>C</math> at <math>(0,0)</math> and <math>(7,1)</math>, respectively. What is its area?
 +
 
 +
<math>
 +
\mathrm{(A)}\ 20\sqrt {3}
 +
\qquad
 +
\mathrm{(B)}\ 22\sqrt {3}
 +
\qquad
 +
\mathrm{(C)}\ 25\sqrt {3}
 +
\qquad
 +
\mathrm{(D)}\ 27\sqrt {3}
 +
\qquad
 +
\mathrm{(E)}\ 50
 +
</math>
  
 
[[2006 AMC 12B Problems/Problem 16|Solution]]
 
[[2006 AMC 12B Problems/Problem 16|Solution]]
  
 
== Problem 17 ==
 
== Problem 17 ==
−
{{problem}}
+
 
 +
For a particular peculiar pair of dice, the probabilities of rolling <math>1</math>, <math>2</math>, <math>3</math>, <math>4</math>, <math>5</math> and <math>6</math> on each die are in the ratio <math>1:2:3:4:5:6</math>. What is the probability of rolling a total of <math>7</math> on the two dice?
 +
 
 +
<math>
 +
\mathrm{(A)}\ \frac 4{63}
 +
\qquad
 +
\mathrm{(B)}\ \frac 18
 +
\qquad
 +
\mathrm{(C)}\ \frac 8{63}
 +
\qquad
 +
\mathrm{(D)}\ \frac 16
 +
\qquad
 +
\mathrm{(E)}\ \frac 27
 +
</math>
  
 
[[2006 AMC 12B Problems/Problem 17|Solution]]
 
[[2006 AMC 12B Problems/Problem 17|Solution]]
  
 
== Problem 18 ==
 
== Problem 18 ==
−
{{problem}}
+
An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?
 +
 
 +
<math>
 +
\mathrm{(A)}\ 120
 +
\qquad
 +
\mathrm{(B)}\ 121
 +
\qquad
 +
\mathrm{(C)}\ 221
 +
\qquad
 +
\mathrm{(D)}\ 230
 +
\qquad
 +
\mathrm{(E)}\ 231
 +
</math>
  
 
[[2006 AMC 12B Problems/Problem 18|Solution]]
 
[[2006 AMC 12B Problems/Problem 18|Solution]]
  
 
== Problem 19 ==
 
== Problem 19 ==
−
{{problem}}
+
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?
 +
 
 +
<math>
 +
\mathrm{(A)}\ 4
 +
\qquad
 +
\mathrm{(B)}\ 5
 +
\qquad
 +
\mathrm{(C)}\ 6
 +
\qquad
 +
\mathrm{(D)}\ 7
 +
\qquad
 +
\mathrm{(E)}\ 8
 +
</math>
  
 
[[2006 AMC 12B Problems/Problem 19|Solution]]
 
[[2006 AMC 12B Problems/Problem 19|Solution]]
  
 
== Problem 20 ==
 
== Problem 20 ==
−
{{problem}}
+
Let <math>x</math> be chosen at random from the interval <math>(0,1)</math>. What is the probability that
 +
<math>\lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0</math>?
 +
Here <math>\lfloor x\rfloor</math> denotes the greatest integer that is less than or equal to <math>x</math>.
 +
 
 +
 
 +
<math>
 +
\mathrm{(A)}\ \frac 18
 +
\qquad
 +
\mathrm{(B)}\ \frac 3{20}
 +
\qquad
 +
\mathrm{(C)}\ \frac 16
 +
\qquad
 +
\mathrm{(D)}\ \frac 15
 +
\qquad
 +
\mathrm{(E)}\ \frac 14
 +
</math>
  
 
[[2006 AMC 12B Problems/Problem 20|Solution]]
 
[[2006 AMC 12B Problems/Problem 20|Solution]]
  
 
== Problem 21 ==
 
== Problem 21 ==
−
{{problem}}
+
Rectangle <math>ABCD</math> has area <math>2006</math>.  An ellipse with area <math>2006\pi</math> passes through <math>A</math> and <math>C</math> and has foci at <math>B</math> and <math>D</math>.  What is the perimeter of the rectangle? (The area of an ellipse is <math>ab\pi</math> where <math>2a</math> and <math>2b</math> are the lengths of the axes.)
 +
 
 +
<math>
 +
\mathrm{(A)}\ \frac {16\sqrt {2006}}{\pi}
 +
\qquad
 +
\mathrm{(B)}\ \frac {1003}4
 +
\qquad
 +
\mathrm{(C)}\ 8\sqrt {1003}
 +
\qquad
 +
\mathrm{(D)}\ 6\sqrt {2006}
 +
\qquad
 +
\mathrm{(E)}\ \frac {32\sqrt {1003}}\pi
 +
</math>
  
 
[[2006 AMC 12B Problems/Problem 21|Solution]]
 
[[2006 AMC 12B Problems/Problem 21|Solution]]
  
 
== Problem 22 ==
 
== Problem 22 ==
−
{{problem}}
+
Suppose <math>a</math>, <math>b</math> and <math>c</math> are positive integers with <math>a+b+c=2006</math>, and <math>a!b!c!=m\cdot 10^n</math>, where <math>m</math> and <math>n</math> are integers and <math>m</math> is not divisible by <math>10</math>. What is the smallest possible value of <math>n</math>?
 +
 
 +
<math>
 +
\mathrm{(A)}\ 489
 +
\qquad
 +
\mathrm{(B)}\ 492
 +
\qquad
 +
\mathrm{(C)}\ 495
 +
\qquad
 +
\mathrm{(D)}\ 498
 +
\qquad
 +
\mathrm{(E)}\ 501
 +
</math>
  
 
[[2006 AMC 12B Problems/Problem 22|Solution]]
 
[[2006 AMC 12B Problems/Problem 22|Solution]]
  
 
== Problem 23 ==
 
== Problem 23 ==
−
{{problem}}
+
Isosceles <math>\triangle ABC</math> has a right angle at <math>C</math>.  Point <math>P</math> is inside <math>\triangle ABC</math>, such that <math>PA=11</math>, <math>PB=7</math>, and <math>PC=6</math>. Legs <math>\overline{AC}</math> and <math>\overline{BC}</math> have length <math>s=\sqrt{a+b\sqrt{2}}</math>, where <math>a</math> and <math>b</math> are positive integers.  What is <math>a+b</math>?
 +
 
 +
<asy>
 +
pathpen = linewidth(0.7); pointpen = black;
 +
pen f = fontsize(10);
 +
size(5cm);
 +
pair B = (0,sqrt(85+42*sqrt(2)));
 +
pair A = (B.y,0);
 +
pair C = (0,0);
 +
pair P = IP(arc(B,7,180,360),arc(C,6,0,90));
 +
D(A--B--C--cycle);
 +
D(P--A);
 +
D(P--B);
 +
D(P--C);
 +
MP("A",D(A),plain.E,f);
 +
MP("B",D(B),plain.N,f);
 +
MP("C",D(C),plain.SW,f);
 +
MP("P",D(P),plain.NE,f);
 +
</asy>
 +
 
 +
<math>
 +
\mathrm{(A)}\ 85
 +
\qquad
 +
\mathrm{(B)}\ 91
 +
\qquad
 +
\mathrm{(C)}\ 108
 +
\qquad
 +
\mathrm{(D)}\ 121
 +
\qquad
 +
\mathrm{(E)}\ 127
 +
</math>
 +
 
  
 
[[2006 AMC 12B Problems/Problem 23|Solution]]
 
[[2006 AMC 12B Problems/Problem 23|Solution]]
  
 
== Problem 24 ==
 
== Problem 24 ==
−
{{problem}}
+
 
 +
Let <math>S</math> be the set of all points <math>(x,y)</math> in the coordinate plane such that <math>0\leq x\leq \frac\pi 2</math> and <math>0\leq y\leq \frac\pi 2</math>. What is the area of the subset of <math>S</math> for which <math>\sin^2 x - \sin x\sin y + \sin^2 y\le \frac 34</math>?
 +
 
 +
<math>
 +
\mathrm{(A)}\ \frac {\pi^2}9
 +
\qquad
 +
\mathrm{(B)}\ \frac {\pi^2}8
 +
\qquad
 +
\mathrm{(C)}\ \frac {\pi^2}6
 +
\qquad
 +
\mathrm{(D)}\ \frac {3\pi^2}{16}
 +
\qquad
 +
\mathrm{(E)}\ \frac {2\pi^2}9
 +
</math>
  
 
[[2006 AMC 12B Problems/Problem 24|Solution]]
 
[[2006 AMC 12B Problems/Problem 24|Solution]]
  
 
== Problem 25 ==
 
== Problem 25 ==
−
{{problem}}
+
 
 +
A sequence <math>a_1,a_2,\dots</math> of non-negative integers is defined by the rule <math>a_{n+2}=|a_{n+1}-a_n|</math> for <math>n\geq 1</math>. If <math>a_1=999</math>, <math>a_2<999</math> and <math>a_{2006}=1</math>, how many different values of <math>a_2</math> are possible?
 +
 
 +
<math>
 +
\mathrm{(A)}\ 165
 +
\qquad
 +
\mathrm{(B)}\ 324
 +
\qquad
 +
\mathrm{(C)}\ 495
 +
\qquad
 +
\mathrm{(D)}\ 499
 +
\qquad
 +
\mathrm{(E)}\ 660
 +
</math>
  
 
[[2006 AMC 12B Problems/Problem 25|Solution]]
 
[[2006 AMC 12B Problems/Problem 25|Solution]]
  
 
== See also ==
 
== See also ==
 +
 +
{{AMC12 box|year=2006|ab=B|before=[[2006 AMC 12A Problems]]|after=[[2007 AMC 12A Problems]]}}
 +
 
* [[AMC 12]]
 
* [[AMC 12]]
 
* [[AMC 12 Problems and Solutions]]
 
* [[AMC 12 Problems and Solutions]]
Line 176: Line 383:
 
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=143 2006 AMC B Math Jam Transcript]
 
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=143 2006 AMC B Math Jam Transcript]
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]
 +
{{MAA Notice}}

Latest revision as of 11:36, 4 July 2023

2006 AMC 12B (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

What is $( - 1)^1 + ( - 1)^2 + \cdots + ( - 1)^{2006}$?

$\text {(A) } - 2006 \qquad \text {(B) } - 1 \qquad \text {(C) } 0 \qquad \text {(D) } 1 \qquad \text {(E) } 2006$

Solution

Problem 2

For real numbers $x$ and $y$, define $x\spadesuit y = (x + y)(x - y)$. What is $3\spadesuit(4\spadesuit 5)$?

$\text {(A) } - 72 \qquad \text {(B) } - 27 \qquad \text {(C) } - 24 \qquad \text {(D) } 24 \qquad \text {(E) } 72$

Solution

Problem 3

A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?

$\text {(A) } 10 \qquad \text {(B) } 14 \qquad \text {(C) } 17 \qquad \text {(D) } 20 \qquad \text {(E) } 24$

Solution

Problem 4

Mary is about to pay for five items at the grocery store. The prices of the items are $\textdollar7.99$, $\textdollar4.99$, $\textdollar2.99$, $\textdollar1.99$, and $\textdollar0.99$. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the $\textdollar20.00$ that she will receive in change?

$\text {(A) } 5 \qquad \text {(B) } 10 \qquad \text {(C) } 15 \qquad \text {(D) } 20 \qquad \text {(E) } 25$

Solution

Problem 5

John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John?

$\text {(A) } 30 \qquad \text {(B) } 50 \qquad \text {(C) } 60 \qquad \text {(D) } 90 \qquad \text {(E) } 120$

Solution

Problem 6

Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?

$\text {(A) } 129 \qquad \text {(B) } 137 \qquad \text {(C) } 174 \qquad \text {(D) } 223 \qquad \text {(E) } 411$

Solution

Problem 7

Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible?

$\text {(A) } 4 \qquad \text {(B) } 12 \qquad \text {(C) } 16 \qquad \text {(D) } 24 \qquad \text {(E) } 48$

Solution

Problem 8

The lines $x = \frac 14y + a$ and $y = \frac 14x + b$ intersect at the point $(1,2)$. What is $a + b$?

$\text {(A) } 0 \qquad \text {(B) } \frac 34 \qquad \text {(C) } 1 \qquad \text {(D) } 2 \qquad \text {(E) } \frac 94$

Solution

Problem 9

How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?

$\text {(A) } 21 \qquad \text {(B) } 34 \qquad \text {(C) } 51 \qquad \text {(D) } 72 \qquad \text {(E) } 150$

Solution

Problem 10

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?

$\text {(A) } 43 \qquad \text {(B) } 44 \qquad \text {(C) } 45 \qquad \text {(D) } 46 \qquad \text {(E) } 47$

Solution

Problem 11

Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?

$\text {(A) } \frac 67 \qquad \text {(B) } \frac {13}{14} \qquad \text {(C) } 1 \qquad \text {(D) } \frac {14}{13} \qquad \text {(E) } \frac 76$

Solution

Problem 12

The parabola $y=ax^2+bx+c$ has vertex $(p,p)$ and $y$-intercept $(0,-p)$, where $p\ne 0$. What is $b$?

$\text {(A) } -p \qquad \text {(B) } 0 \qquad \text {(C) } 2 \qquad \text {(D) } 4 \qquad \text {(E) } p$

Solution

Problem 13

Rhombus $ABCD$ is similar to rhombus $BFDE$. The area of rhombus $ABCD$ is 24, and $\angle BAD = 60^\circ$. What is the area of rhombus $BFDE$?

[asy] defaultpen(linewidth(0.7)+fontsize(11)); unitsize(2cm); pair A=origin, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/sqrt(3)); pair point=(3/2, sqrt(3)/2); draw(B--C--D--A--B--F--D--E--B); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); [/asy]

$\textrm{(A) } 6 \qquad \textrm{(B) } 4\sqrt {3} \qquad \textrm{(C) } 8 \qquad \textrm{(D) } 9 \qquad \textrm{(E) } 6\sqrt {3}$

Solution

Problem 14

Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at $4$ cents per glob and $J$ blobs of jam at $5$ cents per blob. The cost of the peanut butter and jam to make all the sandwiches is $\textdollar 2.53$. Assume that $B$, $J$ and $N$ are all positive integers with $N>1$. What is the cost of the jam Elmo uses to make the sandwiches?

$\mathrm{(A)}\ 1.05 \qquad \mathrm{(B)}\ 1.25 \qquad \mathrm{(C)}\ 1.45 \qquad \mathrm{(D)}\ 1.65 \qquad \mathrm{(E)}\ 1.85$

Solution

Problem 15

Circles with centers $O$ and $P$ have radii 2 and 4, respectively, and are externally tangent. Points $A$ and $B$ are on the circle centered at $O$, and points $C$ and $D$ are on the circle centered at $P$, such that $\overline{AD}$ and $\overline{BC}$ are common external tangents to the circles. What is the area of hexagon $AOBCPD$?

[asy] unitsize(0.4 cm); defaultpen(linewidth(0.7) + fontsize(11)); pair A, B, C, D; pair[] O; O[1] = (6,0); O[2] = (12,0); A = (32/6,8*sqrt(2)/6); B = (32/6,-8*sqrt(2)/6); C = 2*B; D = 2*A; draw(Circle(O[1],2)); draw(Circle(O[2],4)); draw((0.7*A)--(1.2*D)); draw((0.7*B)--(1.2*C)); draw(O[1]--O[2]); draw(A--O[1]); draw(B--O[1]); draw(C--O[2]); draw(D--O[2]); label("$A$", A, NW); label("$B$", B, SW); label("$C$", C, SW); label("$D$", D, NW); dot("$O$", O[1], SE); dot("$P$", O[2], SE); label("$2$", (A + O[1])/2, E); label("$4$", (D + O[2])/2, E);[/asy]

$\textbf{(A) } 18\sqrt {3} \qquad \textbf{(B) } 24\sqrt {2} \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 24\sqrt {3} \qquad \textbf{(E) } 32\sqrt {2}$

Solution

Problem 16

Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$, respectively. What is its area?

$\mathrm{(A)}\ 20\sqrt {3} \qquad \mathrm{(B)}\ 22\sqrt {3} \qquad \mathrm{(C)}\ 25\sqrt {3} \qquad \mathrm{(D)}\ 27\sqrt {3} \qquad \mathrm{(E)}\ 50$

Solution

Problem 17

For a particular peculiar pair of dice, the probabilities of rolling $1$, $2$, $3$, $4$, $5$ and $6$ on each die are in the ratio $1:2:3:4:5:6$. What is the probability of rolling a total of $7$ on the two dice?

$\mathrm{(A)}\ \frac 4{63} \qquad \mathrm{(B)}\ \frac 18 \qquad \mathrm{(C)}\ \frac 8{63} \qquad \mathrm{(D)}\ \frac 16 \qquad \mathrm{(E)}\ \frac 27$

Solution

Problem 18

An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?

$\mathrm{(A)}\ 120 \qquad \mathrm{(B)}\ 121 \qquad \mathrm{(C)}\ 221 \qquad \mathrm{(D)}\ 230 \qquad \mathrm{(E)}\ 231$

Solution

Problem 19

Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?

$\mathrm{(A)}\ 4 \qquad \mathrm{(B)}\ 5 \qquad \mathrm{(C)}\ 6 \qquad \mathrm{(D)}\ 7 \qquad \mathrm{(E)}\ 8$

Solution

Problem 20

Let $x$ be chosen at random from the interval $(0,1)$. What is the probability that $\lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0$? Here $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.


$\mathrm{(A)}\ \frac 18 \qquad \mathrm{(B)}\ \frac 3{20} \qquad \mathrm{(C)}\ \frac 16 \qquad \mathrm{(D)}\ \frac 15 \qquad \mathrm{(E)}\ \frac 14$

Solution

Problem 21

Rectangle $ABCD$ has area $2006$. An ellipse with area $2006\pi$ passes through $A$ and $C$ and has foci at $B$ and $D$. What is the perimeter of the rectangle? (The area of an ellipse is $ab\pi$ where $2a$ and $2b$ are the lengths of the axes.)

$\mathrm{(A)}\ \frac {16\sqrt {2006}}{\pi} \qquad \mathrm{(B)}\ \frac {1003}4 \qquad \mathrm{(C)}\ 8\sqrt {1003} \qquad \mathrm{(D)}\ 6\sqrt {2006} \qquad \mathrm{(E)}\ \frac {32\sqrt {1003}}\pi$

Solution

Problem 22

Suppose $a$, $b$ and $c$ are positive integers with $a+b+c=2006$, and $a!b!c!=m\cdot 10^n$, where $m$ and $n$ are integers and $m$ is not divisible by $10$. What is the smallest possible value of $n$?

$\mathrm{(A)}\ 489 \qquad \mathrm{(B)}\ 492 \qquad \mathrm{(C)}\ 495 \qquad \mathrm{(D)}\ 498 \qquad \mathrm{(E)}\ 501$

Solution

Problem 23

Isosceles $\triangle ABC$ has a right angle at $C$. Point $P$ is inside $\triangle ABC$, such that $PA=11$, $PB=7$, and $PC=6$. Legs $\overline{AC}$ and $\overline{BC}$ have length $s=\sqrt{a+b\sqrt{2}}$, where $a$ and $b$ are positive integers. What is $a+b$?

[asy] pathpen = linewidth(0.7); pointpen = black; pen f = fontsize(10); size(5cm); pair B = (0,sqrt(85+42*sqrt(2))); pair A = (B.y,0); pair C = (0,0); pair P = IP(arc(B,7,180,360),arc(C,6,0,90)); D(A--B--C--cycle); D(P--A); D(P--B); D(P--C); MP("A",D(A),plain.E,f); MP("B",D(B),plain.N,f); MP("C",D(C),plain.SW,f); MP("P",D(P),plain.NE,f); [/asy]

$\mathrm{(A)}\ 85 \qquad \mathrm{(B)}\ 91 \qquad \mathrm{(C)}\ 108 \qquad \mathrm{(D)}\ 121 \qquad \mathrm{(E)}\ 127$


Solution

Problem 24

Let $S$ be the set of all points $(x,y)$ in the coordinate plane such that $0\leq x\leq \frac\pi 2$ and $0\leq y\leq \frac\pi 2$. What is the area of the subset of $S$ for which $\sin^2 x - \sin x\sin y + \sin^2 y\le \frac 34$?

$\mathrm{(A)}\ \frac {\pi^2}9 \qquad \mathrm{(B)}\ \frac {\pi^2}8 \qquad \mathrm{(C)}\ \frac {\pi^2}6 \qquad \mathrm{(D)}\ \frac {3\pi^2}{16} \qquad \mathrm{(E)}\ \frac {2\pi^2}9$

Solution

Problem 25

A sequence $a_1,a_2,\dots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\geq 1$. If $a_1=999$, $a_2<999$ and $a_{2006}=1$, how many different values of $a_2$ are possible?

$\mathrm{(A)}\ 165 \qquad \mathrm{(B)}\ 324 \qquad \mathrm{(C)}\ 495 \qquad \mathrm{(D)}\ 499 \qquad \mathrm{(E)}\ 660$

Solution

See also

2006 AMC 12B (Problems • Answer Key • Resources)
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2006 AMC 12A Problems 		Followed by
2007 AMC 12A Problems
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All AMC 12 Problems and Solutions

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