# 2009 AMC 12A Problems

## 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 ?