# 2012 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

A bug crawls along a number line, starting at -2. It crawls to -6, then turns around and crawls to 5. How many units does the bug crawl altogether?

## Problem 2

Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes?

## 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

Let , , and be positive integers with such that and

What is ?

## Problem 22

## Problem 23

Let be the square one of whose diagonals has endpoints and . A point is chosen uniformly at random over all pairs of real numbers and such that and . Let be a translated copy of centered at . What is the probability that the square region determined by contains exactly two points with integer coefficients in its interior?

## Problem 24

Let be the sequence of real numbers defined by , and in general,

\[a_k=\left\{\array{c}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if k is odd,}\\(0.\underbrace{20101\cdots 01011}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if k is even.}\] (Error compiling LaTeX. ! Missing $ inserted.)

Rearranging the numbers in the sequence in decreasing order produces a new sequence . What is the sum of all integers , , such that

$\textbf{(A)}\ 671\qquad\textbf{(B)}\ 1006\qquad\textbf{(C)}\ 1341\qquad\textbf{(D)}\ 2011\qquad\textbf{(E)}\2012$ (Error compiling LaTeX. ! Undefined control sequence.)

## Problem 25

Let where denotes the fractional part of . The number is the smallest positive integer such that the equation has at least real solutions. What is ? **Note:** the fractional part of is a real number such that and is an integer.

$\textbf{(A)}\ 30\qquad\textbf{(B)}\ 31\qquad\textbf{(C)}\ 32\qquad\textbf{(D)}\ 62\qquad\textbf{(E)}\64$ (Error compiling LaTeX. ! Undefined control sequence.)