2019AMC10 B卷真题及答案

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

In a high school with

students,

of the seniors play a musical instrument, while

of the non-seniors do not play a musical instrument. In all,

of the students do not play a musical instrument. How many non-seniors play a musical instrument?

Problem 4

All lines with equation

such that

form an arithmetic progression pass through a common point. What are the coordinates of that point?

Problem 5

Triangle

lies in the first quadrant. Points

, and

are reflected across the line

to points

, and

,respectively. Assume that none of the vertices of the triangle lie on the line

. Which of the following statements is notalways true?

Triangle

lies in the first quadrant.

Triangles

and

have the same area.

The slope of line

The slopes of lines

and

are the same.

Lines

and

are perpendicular to each other.

Problem 6

There is a real

such that

. What is the sum of the digits of

Problem 7

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 8

The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length 2 and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region?

[Asymptote diagram needed]

Problem 9

The function

is defined by

for all real numbers

, where

denotes the greatest integer less than or equal to the real number

. What is the range of

Problem 10

In a given plane, points

and

are

units apart. How many points

are there in the plane such that the perimeter of

units and the area of

square units?

Problem 11

Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar 1 the ratio of blue to green marbles is 9:1, and the ratio of blue to green marbles in Jar 2 is 8:1. There are 95 green marbles in all. How many more blue marbles are in Jar 1 than in Jar 2?

Problem 12

What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than

Problem 13

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 14

The base-ten representation for

, where

, and

denote digits that are not given. What is

Problem 15

Two right triangles,

and

, have areas of 1 and 2, respectively. One side length of one triangle is congruent to a different side length in the other, and another side length of the first triangle is congruent to yet another side length in the other. What is the product of the third side lengths of

and

A) 28/3 B) 10 C) 32/3 D)34/3 E) 12

Problem 16

with a right angle at

point

lies in the interior of

and point

lies in the interior of

so that

and the ratio

What is the ratio

Problem 17

A red ball and a green ball are randomly and independently tossed into bins numbered with positive integers so that for each ball, the probability that it is tossed into bin

for

What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?

Problem 18

Henry decides one morning to do a workout, and he walks

of the way from his home to his gym. The gym is

kilometers away from Henry’s home. At that point, he changes his mind and walks

of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks

of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked

of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point

kilometers from home and a point

kilometers from home. What is

Problem 19

Let

be the set of all positive integer divisors of

How many numbers are the product of two distinct elements of

Problem 20

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

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 21

Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head?

Problem 22

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 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.)