# Difference between revisions of "2009 AMC 12A Problems"

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== Problem 1 == | == Problem 1 == | ||

+ | Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took <math>h</math> hours and <math>m</math> minutes, with <math>0 < m < 60</math>, what is <math>h + m</math>? | ||

+ | |||

+ | <math>\textbf{(A)}\ 46 \qquad \textbf{(B)}\ 47 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 53 \qquad \textbf{(E)}\ 54</math> | ||

[[2009 AMC 12A Problems/Problem 1|Solution]] | [[2009 AMC 12A Problems/Problem 1|Solution]] | ||

== Problem 2 == | == Problem 2 == | ||

+ | Which of the following is equal to <math>1 + \frac {1}{1 + \frac {1}{1 + 1}}</math>? | ||

+ | |||

+ | <math>\textbf{(A)}\ \frac {5}{4} \qquad \textbf{(B)}\ \frac {3}{2} \qquad \textbf{(C)}\ \frac {5}{3} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3</math> | ||

[[2009 AMC 12A Problems/Problem 2|Solution]] | [[2009 AMC 12A Problems/Problem 2|Solution]] | ||

== Problem 3 == | == Problem 3 == | ||

+ | What number is one third of the way from <math>\frac14</math> to <math>\frac34</math>? | ||

+ | |||

+ | <math>\textbf{(A)}\ \frac {1}{3} \qquad \textbf{(B)}\ \frac {5}{12} \qquad \textbf{(C)}\ \frac {1}{2} \qquad \textbf{(D)}\ \frac {7}{12} \qquad \textbf{(E)}\ \frac {2}{3}</math> | ||

[[2009 AMC 12A Problems/Problem 3|Solution]] | [[2009 AMC 12A Problems/Problem 3|Solution]] | ||

== Problem 4 == | == Problem 4 == | ||

+ | Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could [i]not[/i] be the total value of the four coins, in cents? | ||

+ | |||

+ | <math>\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 55</math> | ||

[[2009 AMC 12A Problems/Problem 4|Solution]] | [[2009 AMC 12A Problems/Problem 4|Solution]] | ||

== Problem 5 == | == Problem 5 == | ||

+ | One dimension of a cube is increased by <math>1</math>, another is decreased by <math>1</math>, and the third is left unchanged. The volume of the new rectangular solid is <math>5</math> less than that of the cube. What was the volume of the cube? | ||

+ | |||

+ | <math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 125 \qquad \textbf{(E)}\ 216</math> | ||

[[2009 AMC 12A Problems/Problem 5|Solution]] | [[2009 AMC 12A Problems/Problem 5|Solution]] | ||

== Problem 6 == | == Problem 6 == | ||

+ | Suppose that <math>P = 2^m</math> and <math>Q = 3^n</math>. Which of the following is equal to <math>12^{mn}</math> for every pair of integers <math>(m,n)</math>? | ||

+ | |||

+ | <math>\textbf{(A)}\ P^2Q \qquad \textbf{(B)}\ P^nQ^m \qquad \textbf{(C)}\ P^nQ^{2m} \qquad \textbf{(D)}\ P^{2m}Q^n \qquad \textbf{(E)}\ P^{2n}Q^m</math> | ||

[[2009 AMC 12A Problems/Problem 6|Solution]] | [[2009 AMC 12A Problems/Problem 6|Solution]] | ||

== Problem 7 == | == Problem 7 == | ||

+ | The first three terms of an arithmetic sequence are <math>2x - 3</math>, <math>5x - 11</math>, and <math>3x + 1</math> respectively. The <math>n</math>th term of the sequence is <math>2009</math>. What is <math>n</math>? | ||

+ | |||

+ | <math>\textbf{(A)}\ 255 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 1506 \qquad \textbf{(E)}\ 8037</math> | ||

[[2009 AMC 12A Problems/Problem 7|Solution]] | [[2009 AMC 12A Problems/Problem 7|Solution]] | ||

== Problem 8 == | == Problem 8 == | ||

+ | Four congruent rectangles are placed as shown. The area of the outer square is <math>4</math> times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side? | ||

+ | <center><asy> | ||

+ | unitsize(6mm); | ||

+ | defaultpen(linewidth(.8pt)); | ||

+ | |||

+ | path p=(1,1)--(-2,1)--(-2,2)--(1,2); | ||

+ | draw(p); | ||

+ | draw(rotate(90)*p); | ||

+ | draw(rotate(180)*p); | ||

+ | draw(rotate(270)*p); | ||

+ | </asy></center> | ||

+ | |||

+ | <math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \sqrt {10} \qquad \textbf{(C)}\ 2 + \sqrt2 \qquad \textbf{(D)}\ 2\sqrt3 \qquad \textbf{(E)}\ 4</math> | ||

[[2009 AMC 12A Problems/Problem 8|Solution]] | [[2009 AMC 12A Problems/Problem 8|Solution]] | ||

Line 36: | Line 70: | ||

== Problem 10 == | == Problem 10 == | ||

+ | In quadrilateral <math>ABCD</math>, <math>AB = 5</math>, <math>BC = 17</math>, <math>CD = 5</math>, <math>DA = 9</math>, and <math>BD</math> is an integer. What is <math>BD</math>? | ||

+ | <center><asy> | ||

+ | unitsize(4mm); | ||

+ | defaultpen(linewidth(.8pt)+fontsize(8pt)); | ||

+ | dotfactor=4; | ||

+ | |||

+ | pair C=(0,0), B=(17,0); | ||

+ | pair D=intersectionpoints(Circle(C,5),Circle(B,13))[0]; | ||

+ | pair A=intersectionpoints(Circle(D,9),Circle(B,5))[0]; | ||

+ | pair[] dotted={A,B,C,D}; | ||

+ | |||

+ | draw(D--A--B--C--D--B); | ||

+ | dot(dotted); | ||

+ | label("$D$",D,NW); | ||

+ | label("$C$",C,W); | ||

+ | label("$B$",B,E); | ||

+ | label("$A$",A,NE); | ||

+ | </asy></center><math>\textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15</math> | ||

[[2009 AMC 12A Problems/Problem 10|Solution]] | [[2009 AMC 12A Problems/Problem 10|Solution]] | ||

== Problem 11 == | == Problem 11 == | ||

+ | The figures <math>F_1</math>, <math>F_2</math>, <math>F_3</math>, and <math>F_4</math> shown are the first in a sequence of figures. For <math>n\ge3</math>, <math>F_n</math> is constructed from <math>F_{n - 1}</math> by surrounding it with a square and placing one more diamond on each side of the new square than <math>F_{n - 1}</math> had on each side of its outside square. For example, figure <math>F_3</math> has <math>13</math> diamonds. How many diamonds are there in figure <math>F_{20}</math>? | ||

+ | <center><asy> | ||

+ | unitsize(3mm); | ||

+ | defaultpen(linewidth(.8pt)+fontsize(8pt)); | ||

+ | |||

+ | path d=(1/2,0)--(0,sqrt(3)/2)--(-1/2,0)--(0,-sqrt(3)/2)--cycle; | ||

+ | marker m=marker(scale(5)*d,Fill); | ||

+ | path f1=(0,0); | ||

+ | path f2=(0,0)--(-1,1)--(1,1)--(1,-1)--(-1,-1); | ||

+ | path[] g2=(-1,1)--(-1,-1)--(0,0)^^(1,-1)--(0,0)--(1,1); | ||

+ | path f3=f2--(-2,-2)--(-2,0)--(-2,2)--(0,2)--(2,2)--(2,0)--(2,-2)--(0,-2); | ||

+ | path[] g3=g2^^(-2,-2)--(0,-2)^^(2,-2)--(1,-1)^^(1,1)--(2,2)^^(-1,1)--(-2,2); | ||

+ | path[] f4=f3^^(-3,-3)--(-3,-1)--(-3,1)--(-3,3)--(-1,3)--(1,3)--(3,3)-- | ||

+ | (3,1)--(3,-1)--(3,-3)--(1,-3)--(-1,-3); | ||

+ | path[] g4=g3^^(-2,-2)--(-3,-3)--(-1,-3)^^(3,-3)--(2,-2)^^(2,2)--(3,3)^^ | ||

+ | (-2,2)--(-3,3); | ||

+ | |||

+ | draw(f1,m); | ||

+ | draw(shift(5,0)*f2,m); | ||

+ | draw(shift(5,0)*g2); | ||

+ | draw(shift(12,0)*f3,m); | ||

+ | draw(shift(12,0)*g3); | ||

+ | draw(shift(21,0)*f4,m); | ||

+ | draw(shift(21,0)*g4); | ||

+ | label("$F_1$",(0,-4)); | ||

+ | label("$F_2$",(5,-4)); | ||

+ | label("$F_3$",(12,-4)); | ||

+ | label("$F_4$",(21,-4)); | ||

+ | </asy></center><math>\textbf{(A)}\ 401 \qquad \textbf{(B)}\ 485 \qquad \textbf{(C)}\ 585 \qquad \textbf{(D)}\ 626 \qquad \textbf{(E)}\ 761</math> | ||

[[2009 AMC 12A Problems/Problem 11|Solution]] | [[2009 AMC 12A Problems/Problem 11|Solution]] | ||

== Problem 12 == | == Problem 12 == | ||

+ | How many positive integers less than <math>1000</math> are <math>6</math> times the sum of their digits? | ||

+ | |||

+ | <math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 12</math> | ||

[[2009 AMC 12A Problems/Problem 12|Solution]] | [[2009 AMC 12A Problems/Problem 12|Solution]] | ||

== Problem 13 == | == Problem 13 == | ||

+ | A ship sails <math>10</math> miles in a straight line from <math>A</math> to <math>B</math>, turns through an angle between <math>45^{\circ}</math> and <math>60^{\circ}</math>, and then sails another <math>20</math> miles to <math>C</math>. Let <math>AC</math> be measured in miles. Which of the following intervals contains <math>AC^2</math>? | ||

+ | <asy> | ||

+ | unitsize(2mm); | ||

+ | defaultpen(linewidth(.8pt)+fontsize(10pt)); | ||

+ | dotfactor=4; | ||

+ | |||

+ | pair B=(0,0), A=(-10,0), C=20*dir(50); | ||

+ | |||

+ | draw(A--B--C); | ||

+ | draw(A--C,linetype("4 4")); | ||

+ | |||

+ | dot(A); | ||

+ | dot(B); | ||

+ | dot(C); | ||

+ | label("$10$",midpoint(A--B),S); | ||

+ | label("$20$",midpoint(B--C),SE); | ||

+ | label("$A$",A,SW); | ||

+ | label("$B$",B,SE); | ||

+ | label("$C$",C,NE); | ||

+ | </asy></center><math>\textbf{(A)}\ [400,500] \qquad \textbf{(B)}\ [500,600] \qquad \textbf{(C)}\ [600,700] \qquad \textbf{(D)}\ [700,800]</math> | ||

+ | <math>\textbf{(E)}\ [800,900]</math> | ||

[[2009 AMC 12A Problems/Problem 13|Solution]] | [[2009 AMC 12A Problems/Problem 13|Solution]] | ||

== Problem 14 == | == Problem 14 == | ||

+ | A triangle has vertices <math>(0,0)</math>, <math>(1,1)</math>, and <math>(6m,0)</math>, and the line <math>y = mx</math> divides the triangle into two triangles of equal area. What is the sum of all possible values of <math>m</math>? | ||

+ | |||

+ | <math>\textbf{(A)} - \!\frac {1}{3} \qquad \textbf{(B)} - \!\frac {1}{6} \qquad \textbf{(C)}\ \frac {1}{6} \qquad \textbf{(D)}\ \frac {1}{3} \qquad \textbf{(E)}\ \frac {1}{2}</math> | ||

[[2009 AMC 12A Problems/Problem 14|Solution]] | [[2009 AMC 12A Problems/Problem 14|Solution]] |

## Revision as of 22:32, 11 February 2009

## 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

## Problem 1

Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took hours and minutes, with , what is ?

## Problem 2

Which of the following is equal to ?

## Problem 3

What number is one third of the way from to ?

## Problem 4

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could [i]not[/i] be the total value of the four coins, in cents?

## Problem 5

One dimension of a cube is increased by , another is decreased by , and the third is left unchanged. The volume of the new rectangular solid is less than that of the cube. What was the volume of the cube?

## Problem 6

Suppose that and . Which of the following is equal to for every pair of integers ?

## Problem 7

The first three terms of an arithmetic sequence are , , and respectively. The th term of the sequence is . What is ?

## Problem 8

Four congruent rectangles are placed as shown. The area of the outer square is times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?

## Problem 9

## Problem 10

In quadrilateral , , , , , and is an integer. What is ?

## Problem 11

The figures , , , and shown are the first in a sequence of figures. For , is constructed from by surrounding it with a square and placing one more diamond on each side of the new square than had on each side of its outside square. For example, figure has diamonds. How many diamonds are there in figure ?

## Problem 12

How many positive integers less than are times the sum of their digits?

## Problem 13

A ship sails miles in a straight line from to , turns through an angle between and , and then sails another miles to . Let be measured in miles. Which of the following intervals contains ?

## Problem 14

A triangle has vertices , , and , and the line divides the triangle into two triangles of equal area. What is the sum of all possible values of ?

## Problem 15

For what value of is ?

Note: here .

## Problem 16

A circle with center is tangent to the positive and -axes and externally tangent to the circle centered at with radius . What is the sum of all possible radii of the circle with center ?

## Problem 17

Let and be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is , and the sum of the second series is . What is ?

## Problem 18

For , let , where there are zeros between the and the . Let be the number of factors of in the prime factorization of . What is the maximum value of ?

## Problem 19

Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon (7 sides). The areas of the two regions were and , respectively. Each polygon had a side length of . Which of the following is true?

## Problem 20

Convex quadrilateral has and . Diagonals and intersect at , , and and have equal areas. What is ?

## Problem 21

Let , where , , and are complex numbers. Suppose that

What is the number of nonreal zeros of ?

## Problem 22

A regular octahedron has side length . A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area , where , , and are positive integers, and are relatively prime, and is not divisible by the square of any prime. What is ?

## Problem 23

Functions and are quadratic, , and the graph of contains the vertex of the graph of . The four -intercepts on the two graphs have -coordinates , , , and , in increasing order, and . The value of is , where , , and are positive integers, and is not divisible by the square of any prime. What is ?

## Problem 24

The *tower function of twos* is defined recursively as follows: and for . Let and . What is the largest integer such that

is defined?

## Problem 25

The first two terms of a sequence are and . For ,

What is ?