Difference between revisions of "2017 AMC 12B Problems"
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==Problem 17== | ==Problem 17== | ||
+ | A coin is biased in such a way that on each toss the probability of heads is <math>\frac{2}{3}</math> and the probability of tails is <math>\frac{1}{3}</math>. The outcomes of the tosses are independent. A player has the choice of playing Game A or Game B. In Game A she tosses the coin three times and wins if all three outcomes are the same. In Game B she tosses the coin four times and wins if both the outcomes of the first and second tosses are the same and the outcomes of the third and fourth tosses are the same. How do the chances of winning Game A compare to the chances of winning Game B? | ||
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+ | <math>\textbf{(A)}</math> The probability of winning Game A is <math>\frac{4}{81}</math> less than the probability of winning Game B. | ||
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+ | <math>\textbf{(B)}</math> The probability of winning Game A is <math>\frac{2}{81}</math> less than the probability of winning Game B. | ||
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+ | <math>\textbf{(C)}</math> The probabilities are the same. | ||
+ | |||
+ | <math>\textbf{(D)}</math> The probability of winning Game A is <math>\frac{2}{81}</math> greater than the probability of winning Game B. | ||
+ | |||
+ | <math>\textbf{(E)}</math> The probability of winning Game A is <math>\frac{4}{81}</math> greater than the probability of winning Game B. | ||
==Problem 18== | ==Problem 18== |
Revision as of 20:12, 16 February 2017
WORK IN PROGRESS
2017 AMC 12B (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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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
Kymbrea's comic book collection currently has comic books in it, and she is adding to her collection at the rate of comic books per month. LaShawn's collection currently has comic books in it, and he is adding to his collection at the rate of comic books per month. After how many months will LaShawn's collection have twice as many comic books as Kymbrea's?
Problem 2
Real numbers , , and satify the inequalities , , and . Which of the following numbers is necessarily positive?
Problem 3
Supposed that and are nonzero real numbers such that . What is the value of ?
Problem 4
Samia set off on her bicycle to visit her friend, traveling at an average speed of kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at kilometers per hour. In all it took her minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
Problem 5
Problem 6
The circle having and as the endpoints of a diameter intersects the -axis at a second point. What is the -coordinate of this point?
Problem 7
The functions and are periodic with least period . What is the least period of the function ?
Problem 8
Problem 9
Problem 10
At Typico High School, of the students like dancing, and the rest dislike it. Of those who like dancing, say that they like it, and the rest say that they dislike it. Of those who dislike dancing, say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?
Problem 11
Call a positive integer if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, , , and are monotonous, but , , and are not. How many monotonous positive integers are there?
Problem 12
What is the sum of the roots of that have a positive real part?
Problem 13
Problem 14
An ice-cream novelty item consists of a cup in the shape of a 4-inch-tall frustum of a right circular cone, with a 2-inch-diameter base at the bottom and a 4-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height 4 inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches?
Problem 15
Let be an equilateral triangle. Extend side beyond to a point so that . Similarly, extend side beyond to a point so that , and extend side beyond to a point so that . What is the ratio of the area of to the area of ?
Problem 16
The number has over positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
Problem 17
A coin is biased in such a way that on each toss the probability of heads is and the probability of tails is . The outcomes of the tosses are independent. A player has the choice of playing Game A or Game B. In Game A she tosses the coin three times and wins if all three outcomes are the same. In Game B she tosses the coin four times and wins if both the outcomes of the first and second tosses are the same and the outcomes of the third and fourth tosses are the same. How do the chances of winning Game A compare to the chances of winning Game B?
The probability of winning Game A is less than the probability of winning Game B.
The probability of winning Game A is less than the probability of winning Game B.
The probabilities are the same.
The probability of winning Game A is greater than the probability of winning Game B.
The probability of winning Game A is greater than the probability of winning Game B.
Problem 18
The diameter of a circle of radius is extended to a point outside the circle so that . Point is chosen so that and line is perpendicular to line . Segment intersects the circle at a point between and . What is the area of ?
Problem 19
Let be the -digit number that is formed by writing the integers from to in order, one after the other. What is the remainder when is divided by ?
Problem 20
Real numbers and are chosen independently and uniformly at random from the interval . What is the probability that , where denotes the greatest integer less than or equal to the real number ?
Problem 21
Last year Isabella took math tests and received different scores, each an integer between and , inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was . What was her score on the sixth test?
Problem 22
Problem 23
Problem 24
Problem 25
See also
2017 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2017 AMC 12A Problems |
Followed by 2018 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.