2019 AMC 12B Problems
2019 AMC 12B (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
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
- 26 See also
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
Which one of the following rigid transformations (isometries) maps the line segment onto the line segment so that the image of is and the image of is
reflection in the -axis
counterclockwise rotation around the origin by
translation by units to the right and units down
reflection in the -axis
clockwise rotation about the origin by
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 pieces of red candy, pieces of green candy, pieces of blue candy, or pieces of purple candy. A piece of purple candy costs 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
Let . What is the value of the sum
Problem 9
For how many integral values of can a triangle of positive area be formed having side lengths ?
Problem 10
The figure below is a map showing cities and roads connecting certain pairs of cities. Paula wishes to travel along exactly of those roads, starting at city and ending at city without traveling along any portion of a road more than once. (Paula is allowed to visit a city more than once.)
How many different routes can Paula take?
Problem 11
How many unordered pairs of edges of a given cube determine a plane?
Problem 12
Right triangle with right angle at is constructed outwards on the hypotenuse of isosceles right triangle with leg length , as shown, so that the two triangles have equal perimeters. What is ?
Problem 13
A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin is for What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?
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
As shown in the figure, line segment is trisected by points and so that Three semicircles of radius and have their diameters on and are tangent to line at and respectively. A circle of radius has its center on The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form where and are positive integers and and are relatively prime. What is ?
Problem 16
There are lily pads in a row numbered to , in that order. There are predators on lily pads and , and a morsel of food on lily pad . Fiona the frog starts on pad , and from any given lily pad, has a chance to hop to the next pad, and an equal chance to jump pads. What is the probability that Fiona reaches pad without landing on either pad or pad ?
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
Raashan, Sylvia, and Ted play the following game. Each starts with . A bell rings every seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives to that player. What is the probability that after the bell has rung times, each player will have ? (For example, Raashan and Ted may each decide to give to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have , Sylvia will have , and Ted will have , and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their to, and the holdings will be the same at the end of the second round.)
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 the area of ?
See also
2019 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2019 AMC 12A Problems |
Followed by 2020 AMC 12A Problems |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.