**Art of Problem Solving textbooks**as a central part of their AMC preparation.

# 2019 AMC 12B 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

Alicia had two containers. The first was full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was full of water. What is the ratio of the volume of the first container to the volume of the second container?

## Problem 2

Consider the statement, "If is not prime, then is prime." Which of the following values of is a counterexample to this statement.

## Problem 3

## Problem 4

A positive integer satisfies the equation . What is the sum of the digits of ?

## Problem 5

Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of ?

## Problem 6

In a given plane, points and are units apart. How many points are there in the plane such that the perimeter of is units and the area of is square units?

## Problem 7

What is the sum of all real numbers for which the median of the numbers and is equal to the mean of those five numbers?

## Problem 8

## Problem 9

## Problem 10

## Problem 11

How many unordered pairs of edges of a given cube determine a plane?

## Problem 12

## Problem 13

## Problem 14

Let be the set of all positive integer divisors of How many numbers are the product of two distinct elements of

## Problem 15

## Problem 16

There are lily pads in a row numbered 0 to 11, in that order. There are predators on lily pads 3 and 6, and a morsel of food on lily pad 10. Fiona the frog starts on pad 0, and from any given lily pad, has a chance to hop to the next pad, and an equal chance to jump 2 pads. What is the probability that Fiona reaches pad 10 without landing on either pad 3 or pad 6?

## Problem 17

How many nonzero complex numbers have the property that and when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?

## Problem 18

Square pyramid has base which measures cm on a side, and altitude perpendicular to the base which measures cm. Point lies on one third of the way from to point lies on one third of the way from to and point lies on two thirds of the way from to What is the area, in square centimeters, of

## Problem 19

There are lily pads in a row numbered 0 to 11, in that order. There are predators on lily pads 3 and 6, and a morsel of food on lily pad 10. Fiona the frog starts on pad 0, and from any given lily pad, has a chance to hop to the next pad, and an equal chance to jump 2 pads. What is the probability that Fiona reaches pad 10 without landing on either pad 3 or pad 6?

## Problem 20

Points and lie on circle in the plane. Suppose that the tangent lines to at and intersect at a point on the -axis. What is the area of ?

## Problem 21

How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is and the roots are and then the requirement is that .)

## Problem 22

Define a sequence recursively by and for all nonnegative integers Let be the least positive integer such that In which of the following intervals does lie?

## Problem 23

How many sequences of s and s of length are there that begin with a , end with a , contain no two consecutive s, and contain no three consecutive s?

## Problem 24

Let Let denote all points in the complex plane of the form where and What is the area of ?

## Problem 25

Let be a convex quadrilateral with and Suppose that the centroids of and form the vertices of an equilateral triangle. What is the maximum possible value of ?