# Difference between revisions of "2006 AMC 12A Problems"

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Define <math>x\otimes y=x^3-y</math>. What is <math>h\otimes (h\otimes h)</math>? | Define <math>x\otimes y=x^3-y</math>. What is <math>h\otimes (h\otimes h)</math>? | ||

+ | <center><math> \mathrm{(A) \ } -h\qquad \mathrm{(B) \ } 0\qquad \mathrm{(C) \ } h\qquad \mathrm{(D) \ } 2h\qquad \mathrm{(E) \ } h^3</math></center> | ||

[[2006 AMC 12A Problems/Problem 2|Solution]] | [[2006 AMC 12A Problems/Problem 2|Solution]] | ||

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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? | 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? | ||

+ | <center><math> \mathrm{(A) \ } \qquad \mathrm{(B) \ } \qquad \mathrm{(C) \ } \qquad \mathrm{(D) \ } \qquad \mathrm{(E) \ } </math></center> | ||

[[2006 AMC 12A Problems/Problem 3|Solution]] | [[2006 AMC 12A Problems/Problem 3|Solution]] | ||

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

+ | <center><math> \mathrm{(A) \ } \qquad \mathrm{(B) \ } \qquad \mathrm{(C) \ } \qquad \mathrm{(D) \ } \qquad \mathrm{(E) \ } </math></center> | ||

[[2006 AMC 12A Problems/Problem 4|Solution]] | [[2006 AMC 12A Problems/Problem 4|Solution]] | ||

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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 <math>$8</math>, and there was an additional cost of <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? | 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 <math>$8</math>, and there was an additional cost of <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? | ||

+ | <center><math> \mathrm{(A) \ } \qquad \mathrm{(B) \ } \qquad \mathrm{(C) \ } \qquad \mathrm{(D) \ } \qquad \mathrm{(E) \ } </math></center> | ||

[[2006 AMC 12A Problems/Problem 5|Solution]] | [[2006 AMC 12A Problems/Problem 5|Solution]] | ||

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The <math>8\times 18</math> rectangle <math>ABCD</math> 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 <math>y</math>? | The <math>8\times 18</math> rectangle <math>ABCD</math> 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 <math>y</math>? | ||

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[[2006 AMC 12A Problems/Problem 6|Solution]] | [[2006 AMC 12A Problems/Problem 6|Solution]] | ||

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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? | 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? | ||

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[[2006 AMC 12A Problems/Problem 7|Solution]] | [[2006 AMC 12A Problems/Problem 7|Solution]] | ||

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How many sets of two or more consecutive positive integers have a sum of <math>15</math>? | How many sets of two or more consecutive positive integers have a sum of <math>15</math>? | ||

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[[2006 AMC 12A Problems/Problem 8|Solution]] | [[2006 AMC 12A Problems/Problem 8|Solution]] | ||

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Oscar buys <math>13</math> pencils and <math>3</math> erasers for <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? | Oscar buys <math>13</math> pencils and <math>3</math> erasers for <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? | ||

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[[2006 AMC 12A Problems/Problem 9|Solution]] | [[2006 AMC 12A Problems/Problem 9|Solution]] | ||

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For how many real values of <math>x</math> is <math>\sqrt{120-\sqrt{x}}</math> an integer? | For how many real values of <math>x</math> is <math>\sqrt{120-\sqrt{x}}</math> an integer? | ||

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[[2006 AMC 12A Problems/Problem 10|Solution]] | [[2006 AMC 12A Problems/Problem 10|Solution]] | ||

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Which of the following describes the graph of the equation <math>(x+y)^2=x^2+y^2</math>? | Which of the following describes the graph of the equation <math>(x+y)^2=x^2+y^2</math>? | ||

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[[2006 AMC 12A Problems/Problem 11|Solution]] | [[2006 AMC 12A Problems/Problem 11|Solution]] | ||

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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? | 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? | ||

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[[2006 AMC 12A Problems/Problem 12|Solution]] | [[2006 AMC 12A Problems/Problem 12|Solution]] | ||

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

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[[2006 AMC 12A Problems/Problem 13|Solution]] | [[2006 AMC 12A Problems/Problem 13|Solution]] | ||

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Two farmers agree that pigs are worth <math>$300</math> and that goats are worth <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 <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? | Two farmers agree that pigs are worth <math>$300</math> and that goats are worth <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 <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? | ||

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[[2006 AMC 12A Problems/Problem 14|Solution]] | [[2006 AMC 12A Problems/Problem 14|Solution]] | ||

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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>? | 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>? | ||

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[[2006 AMC 12A Problems/Problem 15|Solution]] | [[2006 AMC 12A Problems/Problem 15|Solution]] | ||

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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>? | 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>? | ||

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[[2006 AMC 12A Problems/Problem 16|Solution]] | [[2006 AMC 12A Problems/Problem 16|Solution]] | ||

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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>? | 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>? | ||

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[[2006 AMC 12A Problems/Problem 17|Solution]] | [[2006 AMC 12A Problems/Problem 17|Solution]] | ||

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What is the largest set of real numbers that can be in the domain of <math>f</math>? | What is the largest set of real numbers that can be in the domain of <math>f</math>? | ||

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[[2006 AMC 12A Problems/Problem 18|Solution]] | [[2006 AMC 12A Problems/Problem 18|Solution]] | ||

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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>? | 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>? | ||

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[[2006 AMC 12A Problems/Problem 19|Solution]] | [[2006 AMC 12A Problems/Problem 19|Solution]] | ||

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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? | 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? | ||

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[[2006 AMC 12A Problems/Problem 20|Solution]] | [[2006 AMC 12A Problems/Problem 20|Solution]] | ||

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What is the ratio of the area of <math>S_2</math> to the area of <math>S_1</math>? | What is the ratio of the area of <math>S_2</math> to the area of <math>S_1</math>? | ||

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[[2006 AMC 12A Problems/Problem 21|Solution]] | [[2006 AMC 12A Problems/Problem 21|Solution]] | ||

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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>? | 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>? | ||

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[[2006 AMC 12A Problems/Problem 22|Solution]] | [[2006 AMC 12A Problems/Problem 22|Solution]] | ||

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

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[[2006 AMC 12A Problems/Problem 23|Solution]] | [[2006 AMC 12A Problems/Problem 23|Solution]] | ||

== Problem 24 == | == Problem 24 == | ||

+ | Let <math>S</math> be the set of all points <math>(x,y)</math> in the coordinate plane such that <math>0\le x\le\frac{\pi}{2}</math> and <math>0\le y\le\frac{\pi}{2}</math>. What is the area of the subset of <math>S</math> for which | ||

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+ | <math>\sin^{2}x-\sin x\sin y+\sin^{2}y\le\frac{3}{4}</math>? | ||

+ | |||

+ | |||

[[2006 AMC 12A Problems/Problem 24|Solution]] | [[2006 AMC 12A Problems/Problem 24|Solution]] | ||

== Problem 25 == | == Problem 25 == | ||

+ | A sequence <math>a_{1},a_{2},\ldots</math> of non-negative integers is defined by the rule <math>a_{n+2}=|a_{n+1}-a_{n}|</math> for <math>n\ge 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? | ||

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[[2006 AMC 12A Problems/Problem 25|Solution]] | [[2006 AMC 12A Problems/Problem 25|Solution]] | ||

## Revision as of 22:44, 9 July 2006

## Contents

- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also

## Problem 1

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

## Problem 2

Define . What is ?

## Problem 3

The ratio of Mary's age to Alice's age is . Alice is years old. How old is Mary?

## 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?

## Problem 5

Doug and Dave shared a pizza with equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was , and there was an additional cost of 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?

## Problem 6

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The rectangle 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 ?

## Problem 7

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

## Problem 8

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

## Problem 9

Oscar buys pencils and erasers for . 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?

## Problem 10

For how many real values of is an integer?

## Problem 11

Which of the following describes the graph of the equation ?

## 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?

## Problem 13

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

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

## Problem 14

Two farmers agree that pigs are worth and that goats are worth . 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 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?

## Problem 15

Suppose and . What is the smallest possible positive value of ?

## Problem 16

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

Circles with centers and have radii and , respectively. A common internal tangent intersects the circles at and , respectively. Lines and intersect at , and . What is ?

## Problem 17

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

Square has side length , a circle centered at has radius , and and are both rational. The circle passes through , and lies on . Point lies on the circle, on the same side of as . Segment is tangent to the circle, and . What is ?

## Problem 18

The function has the property that for each real number in its domain, is also in its domain and

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

## Problem 19

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

Circles with centers and have radii and , respectively. The equation of a common external tangent to the circles can be written in the form with . What is ?

## 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?

## Problem 21

Let

and

.

What is the ratio of the area of to the area of ?

## Problem 22

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

## Problem 23

Given a finite sequence of real numbers, let be the sequence

of real numbers. Define and, for each integer , , define . Suppose , and let . If , then what is ?

## Problem 24

Let be the set of all points in the coordinate plane such that and . What is the area of the subset of for which

?

## Problem 25

A sequence of non-negative integers is defined by the rule for . If , and , how many different values of are possible?