2016 AMC 12B Problems
2016 AMC 12B (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
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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
What is the value of when ?
Problem 2
The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of and is closest to which integer?
Problem 3
Let . What is the value of ?
Problem 4
The ratio of the measures of two acute angles is , and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
Problem 5
The War of started with a declaration of war on Thursday, June , . The peace treaty to end the war was signed days later, on December , . On what day of the week was the treaty signed?
Problem 6
All three vertices of lie on the parabola defined by , with at the origin and parallel to the -axis. The area of the triangle is . What is the length of ?
Problem 7
Josh writes the numbers . He marks out , skips the next number , marks out , and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number , skips the next number , marks out , skips , marks out , and so on to the end. Josh continues in this manner until only one number remains. What is that number?
Problem 8
A thin piece of wood of uniform density in the shape of an equilateral triangle with side length inches weighs ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of inches. Which of the following is closest to the weight, in ounces, of the second piece?
Problem 9
Carl decided to fence in his rectangular garden. He bought fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden?
Problem 10
A quadrilateral has vertices , , , and , where and are integers with . The area of is . What is ?
Problem 11
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line , the line and the line
Problem 12
All the numbers are written in a array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to . What is the number in the center?
Problem 13
Alice and Bob live miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is from Alice's position and from Bob's position. Which of the following is closest to the airplane's altitude, in miles?
Problem 14
The sum of an infinite geometric series is a positive number , and the second term in the series is . What is the smallest possible value of
Problem 15
All the numbers are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?
Problem 16
In how many ways can be written as the sum of an increasing sequence of two or more consecutive positive integers?
Problem 17
In shown in the figure, , , , and is an altitude. Points and lie on sides and , respectively, so that and are angle bisectors, intersecting at and , respectively. What is ?
Problem 18
What is the area of the region enclosed by the graph of the equation
Problem 19
Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times?
Problem 20
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won games and lost games; there were no ties. How many sets of three teams were there in which beat , beat , and beat
Problem 21
Let be a unit square. Let be the midpoint of . For let be the intersection of and , and let be the foot of the perpendicular from to . What is
Problem 22
For a certain positive integer less than , the decimal equivalent of is , a repeating decimal of period of , and the decimal equivalent of is , a repeating decimal of period . In which interval does lie?
Problem 23
What is the volume of the region in three-dimensional space defined by the inequalities and ?
Problem 24
There are exactly ordered quadruplets such that and . What is the smallest possible value for ?
Problem 25
The sequence is defined recursively by , , and for . What is the smallest positive integer such that the product is an integer?
See also
2016 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2016 AMC 12A Problems |
Followed by 2017 AMC 12A Problems |
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All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.