Difference between revisions of "2006 AMC 12A Problems"

Revision as of 23:58, 27 April 2008

Problem 1

Sandwiches at Joe's Fast Food cost $$$$ 3 each and sodas cost $$$$ 2 each. How many dollars will it cost to purchase $5$ sandwiches and $8$ sodas?

$\mathrm{(A) \ } 31\qquad \mathrm{(B) \ } 32\qquad \mathrm{(C) \ } 33\qquad \mathrm{(D) \ } 34\qquad \mathrm{(E) \ } 35$

Solution

Problem 2

Define $x\otimes y=x^3-y$ . What is $h\otimes (h\otimes h)$ ?

$\mathrm{(A) \ } -h\qquad \mathrm{(B) \ } 0\qquad \mathrm{(C) \ } h\qquad \mathrm{(D) \ } 2h\qquad \mathrm{(E) \ } h^3$

Solution

Problem 3

The ratio of Mary's age to Alice's age is $3:5$ . Alice is $30$ years old. How old is Mary?

$\mathrm{(A) \ } 15\qquad \mathrm{(B) \ } 18\qquad \mathrm{(C) \ } 20\qquad \mathrm{(D) \ } 24\qquad \mathrm{(E) \ } 50$

Solution

Problem 4

A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?

$\mathrm{(A) \ } 17\qquad \mathrm{(B) \ } 19\qquad \mathrm{(C) \ } 21\qquad \mathrm{(D) \ } 22\qquad \mathrm{(E) \ } 23$

Solution

Problem 5

Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was <dollar/> $8$ , and there was an additional cost of <dollar/> $2$ for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?

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

Solution

Problem 6

The $8\times 18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y$ ?

$\mathrm{(A) \ } 6\qquad \mathrm{(B) \ } 7\qquad \mathrm{(C) \ } 8\qquad \mathrm{(D) \ } 9\qquad \mathrm{(E) \ } 10$

Solution

Problem 7

Mary is $20%$ (Error compiling LaTeX. Unknown error_msg) older than Sally, and Sally is $40%$ (Error compiling LaTeX. Unknown error_msg) younger than Danielle. The sum of their ages is $23.2$ years. How old will Mary be on her next birthday?

$\mathrm{(A) \ } 7\qquad \mathrm{(B) \ } 8\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11$

Solution

Problem 8

How many sets of two or more consecutive positive integers have a sum of $15$ ?

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

Solution

Problem 9

Oscar buys $13$ pencils and $3$ erasers for <dollar/> $1.00$ . A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser?

$\mathrm{(A) \ } 10\qquad \mathrm{(B) \ } 12\qquad \mathrm{(C) \ } 15\qquad \mathrm{(D) \ } 18\qquad \mathrm{(E) \ } 20$

Solution

Problem 10

For how many real values of $x$ is $\sqrt{120-\sqrt{x}}$ an integer?

$\mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11$

Solution

Problem 11

Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$ ?

$\mathrm{(A)}\ \text{the empty set}\qquad\mathrm{(B)}\ \text{one point}\mathrm{(C)}\ \text{two lines}\qquad\mathrm{(D)}\ \text{a circle}\qquad\mathrm{(E)}\ \text{the entire plane}$

Solution

Problem 12

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outisde diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?

$\mathrm{(A) \ } 171\qquad \mathrm{(B) \ } 173\qquad \mathrm{(C) \ } 182\qquad \mathrm{(D) \ } 188\qquad \mathrm{(E) \ } 210$

Solution

Problem 13

The vertices of a $3-4-5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?

$\mathrm{(A) \ } 12\pi\qquad \mathrm{(B) \ } \frac{25\pi}{2}\qquad \mathrm{(C) \ } 13\pi\qquad \mathrm{(D) \ } \frac{27\pi}{2}\qquad \mathrm{(E) \ } 14\pi$

Solution

Problem 14

Two farmers agree that pigs are worth <dollar/> $300$ and that goats are worth <dollar/> $210$ . When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a <dollar/> $390$ debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?

$\mathrm{(A) \ }$ $5 \qquad \mathrm{(B) \ }$ $10 \qquad \mathrm{(C) \ }$ $30 \qquad \mathrm{(D) \ }$ $90 \qquad \mathrm{(E) \ }$ $210$

Solution

Problem 15

Suppose $\cos x=0$ and $\cos (x+z)=1/2$ . What is the smallest possible positive value of $z$ ?

$\mathrm{(A) \ } \frac{\pi}{6}\qquad \mathrm{(B) \ } \frac{\pi}{3}\qquad \mathrm{(C) \ } \frac{\pi}{2}\qquad \mathrm{(D) \ } \frac{5\pi}{6}\qquad \mathrm{(E) \ } \frac{7\pi}{6}$

Solution

Problem 16

Circles with centers $A$ and $B$ have radii $3$ and $8$ , respectively. A common internal tangent intersects the circles at $C$ and $D$ , respectively. Lines $AB$ and $CD$ intersect at $E$ , and $AE=5$ . What is $CD$ ?

$\mathrm{(A) \ } 13\qquad \mathrm{(B) \ } \frac{44}{3}\qquad \mathrm{(C) \ } \sqrt{221}\qquad \mathrm{(D) \ } \sqrt{255}\qquad \mathrm{(E) \ } \frac{55}{3}$

Solution

Problem 17

Square $ABCD$ has side length $s$ , a circle centered at $E$ has radius $r$ , and $r$ and $s$ are both rational. The circle passes through $D$ , and $D$ lies on $\overline{BE}$ . Point $F$ lies on the circle, on the same side of $\overline{BE}$ as $A$ . Segment $AF$ is tangent to the circle, and $AF=\sqrt{9+5\sqrt{2}}$ . What is $r/s$ ?

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

Solution

Problem 18

The function $f$ has the property that for each real number $x$ in its domain, $1/x$ is also in its domain and

$f(x)+f\left(\frac{1}{x}\right)=x$

What is the largest set of real numbers that can be in the domain of $f$ ?

$\mathrm{(A) \ } \{x|x\ne 0\}\qquad \mathrm{(B) \ } \{x|x<0\}\qquad \mathrm{(C) \ } \{x|x>0\}\qquad \mathrm{(D) \ } \{x|x\ne -1\;\mathrm{and}\; x\ne 0\;\mathrm{and}\; x\ne 1\}\qquad \mathrm{(E) \ } \{-1,1\}$

Solution

Problem 19

Circles with centers $(2,4)$ and $(14,9)$ have radii $4$ and $9$ , respectively. The equation of a common external tangent to the circles can be written in the form $y=mx+b$ with $m>0$ . What is $b$ ?

$\mathrm{(A) \ } \frac{908}{199}\qquad \mathrm{(B) \ } \frac{909}{119}\qquad \mathrm{(C) \ } \frac{130}{17}\qquad \mathrm{(D) \ } \frac{911}{119}\qquad \mathrm{(E) \ } \frac{912}{119}$

Solution

Problem 20

A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?

$\mathrm{(A) \ } \frac{1}{2187}\qquad \mathrm{(B) \ } \frac{1}{729}\qquad \mathrm{(C) \ } \frac{2}{243}\qquad \mathrm{(D) \ } \frac{1}{81}\qquad \mathrm{(E) \ } \frac{5}{243}$

Solution

Problem 21

Let

$S_1=\{(x,y)|\log_{10}(1+x^2+y^2)\le 1+\log_{10}(x+y)\}$

and

$S_2=\{(x,y)|\log_{10}(2+x^2+y^2)\le 2+\log_{10}(x+y)\}$ .

What is the ratio of the area of $S_2$ to the area of $S_1$ ?

$\mathrm{(A) \ } 98\qquad \mathrm{(B) \ } 99\qquad \mathrm{(C) \ } 100\qquad \mathrm{(D) \ } 101\qquad \mathrm{(E) \ } 102$

Solution

Problem 22

A circle of radius $r$ is concentric with and outside a regular hexagon of side length $2$ . The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is $1/2$ . What is $r$ ?

$\mathrm{(A) \ } 2\sqrt{2}+2\sqrt{3}\qquad \mathrm{(B) \ } 3\sqrt{3}+\sqrt{2}\qquad \mathrm{(C) \ } 2\sqrt{6}+\sqrt{3}$ $\mathrm{(D) \ } 3\sqrt{2}+\sqrt{6}\qquad \mathrm{(E) \ } 6\sqrt{2}-\sqrt{3}$

Solution

Problem 23

Given a finite sequence $S=(a_1,a_2,\ldots ,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence

$\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\ldots ,\frac{a_{n-1}+a_n}{2}\right)$

of $n-1$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$ , $2\le m\le n-1$ , define $A^m(S)=A(A^{m-1}(S))$ . Suppose $x>0$ , and let $S=(1,x,x^2,\ldots ,x^{100})$ . If $A^{100}(S)=(1/2^{50})$ , then what is $x$ ?

$\mathrm{(A) \ } 1-\frac{\sqrt{2}}{2}\qquad \mathrm{(B) \ } \sqrt{2}-1\qquad \mathrm{(C) \ } \frac{1}{2}\qquad \mathrm{(D) \ } 2-\sqrt{2}\qquad \mathrm{(E) \ } \frac{\sqrt{2}}{2}$

Solution

Problem 24

The expression

$(x+y+z)^{2006}+(x-y-z)^{2006}$

is simplified by expanding it and combining like terms. How many terms are in the simplified expression?

$\mathrm{(A) \ } 6018\qquad \mathrm{(B) \ } 671,676\qquad \mathrm{(C) \ } 1,007,514\qquad \mathrm{(D) \ } 1,008,016\qquad \mathrm{(E) \ } 2,015,028$

Solution

Problem 25

How many non-empty subsets $S$ of $\lbrace 1,2,3,\ldots ,15\rbrace$ have the following two properties?

$(1)$ No two consecutive integers belong to $S$ .

$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$ .

$\mathrm{(A) \ } 277\qquad \mathrm{(B) \ } 311\qquad \mathrm{(C) \ } 376\qquad \mathrm{(D) \ } 377\qquad \mathrm{(E) \ } 405$

Solution

@@ Line 1: / Line 1: @@
 == Problem 1 ==
 Sandwiches at Joe's Fast Food cost <math>\</math>3 each and sodas cost <math>\</math>2 each. How many dollars will it cost to purchase <math>5</math> sandwiches and <math>8</math> sodas?
@@ Line 8: / Line 7: @@
 == Problem 2 ==
 Define <math>x\otimes y=x^3-y</math>. What is <math>h\otimes (h\otimes h)</math>?
@@ Line 16: / Line 14: @@
 == Problem 3 ==
 The ratio of Mary's age to Alice's age is <math>3:5</math>. Alice is <math>30</math> years old. How old is Mary?
@@ Line 24: / Line 21: @@
 == Problem 4 ==
 A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?
@@ Line 32: / Line 28: @@
 == Problem 5 ==
 Doug and Dave shared a pizza with <math>8</math> equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was <dollar/><math>8</math>, and there was an additional cost of <dollar/><math>2</math> for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?
@@ Line 49: / Line 44: @@
 == Problem 7 ==
 Mary is <math>20%</math> older than Sally, and Sally is <math>40%</math> younger than Danielle. The sum of their ages is <math>23.2</math> years. How old will Mary be on her next birthday?
@@ Line 57: / Line 51: @@
 == Problem 8 ==
 How many sets of two or more consecutive positive integers have a sum of <math>15</math>?
@@ Line 65: / Line 58: @@
 == Problem 9 ==
 Oscar buys <math>13</math> pencils and <math>3</math> erasers for <dollar/><math>1.00</math>. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser?
@@ Line 73: / Line 65: @@
 == Problem 10 ==
 For how many real values of <math>x</math> is <math>\sqrt{120-\sqrt{x}}</math> an integer?
@@ Line 81: / Line 72: @@
 == Problem 11 ==
 Which of the following describes the graph of the equation <math>(x+y)^2=x^2+y^2</math>?
-<math> \mathrm{(A) \ } \;\mathrm{the\; empty\; set}\;\qquad \mathrm{(B) \ } \;\mathrm{one\; point}</math><math>\mathrm{(C) \ } \;\mathrm{two\; lines}\;\qquad \mathrm{(D) \ } \;\mathrm{a\; circle}\;\qquad \mathrm{(E) \ } \;\mathrm{the\; entire \; plane}\;</math>
+<math>\mathrm{(A)}\ \text{the empty set}\qquad\mathrm{(B)}\ \text{one point}\mathrm{(C)}\ \text{two lines}\qquad\mathrm{(D)}\ \text{a circle}\qquad\mathrm{(E)}\ \text{the entire plane}</math>
 [[2006 AMC 12A Problems/Problem 11|Solution]]
 == Problem 12 ==
 {{image}}
@@ Line 99: / Line 88: @@
 == Problem 13 ==
 [[Image:2006_AMC_12A_Problem_13.gif]]
@@ Line 109: / Line 97: @@
 == Problem 14 ==
 Two farmers agree that pigs are worth <dollar/><math>300</math> and that goats are worth <dollar/><math>210</math>. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a <dollar/><math>390</math> debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?
@@ Line 117: / Line 104: @@
 == Problem 15 ==
 Suppose <math>\cos x=0</math> and <math>\cos (x+z)=1/2</math>. What is the smallest possible positive value of <math>z</math>?
@@ Line 125: / Line 111: @@
 == Problem 16 ==
 Circles with centers <math>A</math> and <math>B</math> have radii <math>3</math> and <math>8</math>, respectively. A common internal tangent intersects the circles at <math>C</math> and <math>D</math>, respectively. Lines <math>AB</math> and <math>CD</math> intersect at <math>E</math>, and <math>AE=5</math>. What is <math>CD</math>?
-<center>[[Image:2006_AMC12A-16.png]]</center>
+[[Image:2006_AMC12A-16.png|center]]
 <math> \mathrm{(A) \ } 13\qquad \mathrm{(B) \ } \frac{44}{3}\qquad \mathrm{(C) \ } \sqrt{221}\qquad \mathrm{(D) \ } \sqrt{255}\qquad \mathrm{(E) \ }  \frac{55}{3}</math>
@@ Line 135: / Line 120: @@
 == Problem 17 ==
 Square <math>ABCD</math> has side length <math>s</math>, a circle centered at <math>E</math> has radius <math>r</math>, and <math>r</math> and <math>s</math> are both rational. The circle passes through <math>D</math>, and <math>D</math> lies on <math>\overline{BE}</math>. Point <math>F</math> lies on the circle, on the same side of <math>\overline{BE}</math> as <math>A</math>. Segment <math>AF</math> is tangent to the circle, and <math>AF=\sqrt{9+5\sqrt{2}}</math>. What is <math>r/s</math>?
-<center>[[Image:AMC12_2006A_17.png]]</center>
+[[Image:AMC12_2006A_17.png|center]]
 <math> \mathrm{(A) \ } \frac{1}{2}\qquad \mathrm{(B) \ } \frac{5}{9}\qquad \mathrm{(C) \ } \frac{3}{5}\qquad \mathrm{(D) \ } \frac{5}{3}\qquad \mathrm{(E) \ }  \frac{9}{5}</math>
@@ Line 145: / Line 129: @@
 == Problem 18 ==
+The function <math>f</math> has the property that for each real number <math>x</math> in its domain, <math>1/x</math> is also in its domain and
-The function <math>\displaystyle f</math> has the property that for each real number <math>\displaystyle x</math> in its domain, <math>\displaystyle 1/x</math> is also in its domain and
 <math>f(x)+f\left(\frac{1}{x}\right)=x</math>
@@ Line 157: / Line 140: @@
 == Problem 19 ==
 Circles with centers <math>(2,4)</math> and <math>(14,9)</math> have radii <math>4</math> and <math>9</math>, respectively. The equation of a common external tangent to the circles can be written in the form <math>y=mx+b</math> with <math>m>0</math>. What is <math>b</math>?
-<center>[[Image:AMC12_2006A_19.png]]</center>
+[[Image:AMC12_2006A_19.png|center]]
 <math> \mathrm{(A) \ } \frac{908}{199}\qquad \mathrm{(B) \ } \frac{909}{119}\qquad \mathrm{(C) \ } \frac{130}{17}\qquad \mathrm{(D) \ } \frac{911}{119}\qquad \mathrm{(E) \ }  \frac{912}{119}</math>
@@ Line 167: / Line 149: @@
 == Problem 20 ==
 A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?
@@ Line 175: / Line 156: @@
 == Problem 21 ==
 Let
@@ Line 191: / Line 171: @@
 == Problem 22 ==
 A circle of radius <math>r</math> is concentric with and outside a regular hexagon of side length <math>2</math>. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is <math>1/2</math>. What is <math>r</math>?
@@ Line 199: / Line 178: @@
 == Problem 23 ==
 Given a finite sequence <math>S=(a_1,a_2,\ldots ,a_n)</math> of <math>n</math> real numbers, let <math>A(S)</math> be the sequence
@@ Line 211: / Line 189: @@
 == Problem 24 ==
 The expression
@@ Line 223: / Line 200: @@
 == Problem 25 ==
+How many non-empty subsets <math>S</math> of <math>\lbrace 1,2,3,\ldots ,15\rbrace</math> have the following two properties?
-How many non-empty subsets <math>S</math> of <math>\{1,2,3,\ldots ,15\}</math> have the following two properties?
 <math>(1)</math>  No two consecutive integers belong to <math>S</math>.

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Difference between revisions of "2006 AMC 12A Problems"

Revision as of 23:58, 27 April 2008

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See also