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

(create) |
(upload) |
||

Line 56: | Line 56: | ||

== Problem 15 == | == Problem 15 == | ||

+ | For what value of <math>n</math> is <math>i + 2i^2 + 3i^3 + \cdots + ni^n = 48 + 49i</math>? | ||

+ | |||

+ | Note: here <math>i = \sqrt { - 1}</math>. | ||

+ | |||

+ | <math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 49 \qquad \textbf{(D)}\ 97 \qquad \textbf{(E)}\ 98</math> | ||

[[2009 AMC 12A Problems/Problem 15|Solution]] | [[2009 AMC 12A Problems/Problem 15|Solution]] | ||

== Problem 16 == | == Problem 16 == | ||

+ | A circle with center <math>C</math> is tangent to the positive <math>x</math> and <math>y</math>-axes and externally tangent to the circle centered at <math>(3,0)</math> with radius <math>1</math>. What is the sum of all possible radii of the circle with center <math>C</math>? | ||

+ | |||

+ | <math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9</math> | ||

[[2009 AMC 12A Problems/Problem 16|Solution]] | [[2009 AMC 12A Problems/Problem 16|Solution]] | ||

== Problem 17 == | == Problem 17 == | ||

+ | Let <math>a + ar_1 + ar_1^2 + ar_1^3 + \cdots</math> and <math>a + ar_2 + ar_2^2 + ar_2^3 + \cdots</math> be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is <math>r_1</math>, and the sum of the second series is <math>r_2</math>. What is <math>r_1 + r_2</math>? | ||

+ | |||

+ | <math>\textbf{(A)}\ 0\qquad \textbf{(B)}\ \frac {1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \frac {1 + \sqrt {5}}{2}\qquad \textbf{(E)}\ 2</math> | ||

[[2009 AMC 12A Problems/Problem 17|Solution]] | [[2009 AMC 12A Problems/Problem 17|Solution]] | ||

== Problem 18 == | == Problem 18 == | ||

+ | For <math>k > 0</math>, let <math>I_k = 10\ldots 064</math>, where there are <math>k</math> zeros between the <math>1</math> and the <math>6</math>. Let <math>N(k)</math> be the number of factors of <math>2</math> in the prime factorization of <math>I_k</math>. What is the maximum value of <math>N(k)</math>? | ||

+ | |||

+ | <math>\textbf{(A)}\ 6\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ 10</math> | ||

[[2009 AMC 12A Problems/Problem 18|Solution]] | [[2009 AMC 12A Problems/Problem 18|Solution]] | ||

== Problem 19 == | == 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 <math>A</math> and <math>B</math>, respectively. Each polygon had a side length of <math>2</math>. Which of the following is true? | ||

+ | |||

+ | <math>\textbf{(A)}\ A = \frac {25}{49}B\qquad \textbf{(B)}\ A = \frac {5}{7}B\qquad \textbf{(C)}\ A = B\qquad \textbf{(D)}\ A </math> <math>= \frac {7}{5}B\qquad \textbf{(E)}\ A = \frac {49}{25}B</math> | ||

[[2009 AMC 12A Problems/Problem 19|Solution]] | [[2009 AMC 12A Problems/Problem 19|Solution]] | ||

== Problem 20 == | == Problem 20 == | ||

+ | Convex quadrilateral <math>ABCD</math> has <math>AB = 9</math> and <math>CD = 12</math>. Diagonals <math>AC</math> and <math>BD</math> intersect at <math>E</math>, <math>AC = 14</math>, and <math>\triangle AED</math> and <math>\triangle BEC</math> have equal areas. What is <math>AE</math>? | ||

+ | |||

+ | <math>\textbf{(A)}\ \frac {9}{2}\qquad \textbf{(B)}\ \frac {50}{11}\qquad \textbf{(C)}\ \frac {21}{4}\qquad \textbf{(D)}\ \frac {17}{3}\qquad \textbf{(E)}\ 6</math> | ||

[[2009 AMC 12A Problems/Problem 20|Solution]] | [[2009 AMC 12A Problems/Problem 20|Solution]] | ||

== Problem 21 == | == Problem 21 == | ||

+ | Let <math>p(x) = x^3 + ax^2 + bx + c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are complex numbers. Suppose that | ||

+ | |||

+ | <center><cmath>p(2009 + 9002\pi i) = p(2009) = p(9002) = 0</cmath></center> | ||

+ | |||

+ | What is the number of nonreal zeros of <math>x^{12} + ax^8 + bx^4 + c</math>? | ||

+ | |||

+ | <math>\textbf{(A)}\ 4\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 12</math> | ||

[[2009 AMC 12A Problems/Problem 21|Solution]] | [[2009 AMC 12A Problems/Problem 21|Solution]] | ||

== Problem 22 == | == Problem 22 == | ||

+ | A regular octahedron has side length <math>1</math>. 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 <math>\frac {a\sqrt {b}}{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers, <math>a</math> and <math>c</math> are relatively prime, and <math>b</math> is not divisible by the square of any prime. What is <math>a + b + c</math>? | ||

+ | |||

+ | <math>\textbf{(A)}\ 10\qquad \textbf{(B)}\ 11\qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 13\qquad \textbf{(E)}\ 14</math> | ||

[[2009 AMC 12A Problems/Problem 22|Solution]] | [[2009 AMC 12A Problems/Problem 22|Solution]] | ||

== Problem 23 == | == Problem 23 == | ||

+ | Functions <math>f</math> and <math>g</math> are quadratic, <math>g(x) = - f(100 - x)</math>, and the graph of <math>g</math> contains the vertex of the graph of <math>f</math>. The four <math>x</math>-intercepts on the two graphs have <math>x</math>-coordinates <math>x_1</math>, <math>x_2</math>, <math>x_3</math>, and <math>x_4</math>, in increasing order, and <math>x_3 - x_2 = 150</math>. The value of <math>x_4 - x_1</math> is <math>m + n\sqrt p</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, and <math>p</math> is not divisible by the square of any prime. What is <math>m + n + p</math>? | ||

+ | |||

+ | <math>\textbf{(A)}\ 602\qquad \textbf{(B)}\ 652\qquad \textbf{(C)}\ 702\qquad \boxed{\textbf{(D)}\ 752}\qquad \textbf{(E)}\ 802</math> | ||

[[2009 AMC 12A Problems/Problem 23|Solution]] | [[2009 AMC 12A Problems/Problem 23|Solution]] | ||

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

+ | The ''tower function of twos'' is defined recursively as follows: <math>T(1) = 2</math> and <math>T(n + 1) = 2^{T(n)}</math> for <math>n\ge1</math>. Let <math>A = (T(2009))^{T(2009)}</math> and <math>B = (T(2009))^A</math>. What is the largest integer <math>k</math> such that | ||

+ | |||

+ | <center><cmath>\underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}</cmath></center> | ||

+ | |||

+ | is defined? | ||

+ | <math>\textbf{(A)}\ 2009\qquad \textbf{(B)}\ 2010\qquad \textbf{(C)}\ 2011\qquad \textbf{(D)}\ 2012\qquad \textbf{(E)}\ 2013</math> | ||

[[2009 AMC 12A Problems/Problem 24|Solution]] | [[2009 AMC 12A Problems/Problem 24|Solution]] | ||

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

+ | The first two terms of a sequence are <math>a_1 = 1</math> and <math>a_2 = \frac {1}{\sqrt3}</math>. For <math>n\ge1</math>, | ||

+ | |||

+ | <center><cmath>a_{n + 2} = \frac {a_n + a_{n + 1}}{1 - a_na_{n + 1}}.</cmath></center> | ||

+ | |||

+ | What is <math>|a_{2009}|</math>? | ||

+ | |||

+ | <math>\textbf{(A)}\ 0\qquad \textbf{(B)}\ 2 - \sqrt3\qquad \textbf{(C)}\ \frac {1}{\sqrt3}\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 2 + \sqrt3</math> | ||

[[2009 AMC 12A Problems/Problem 25|Solution]] | [[2009 AMC 12A Problems/Problem 25|Solution]] |

## Revision as of 17:57, 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

## 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

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 ?