2016 AMC 12A Problems
2016 AMC 12A (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
Problem 1
What is the value of ?
Problem 2
For what value of does
?
Problem 3
The remainder function can be defined for all real numbers and
with
by
,
where denotes the greatest integer less than or equal to
. What is the value of
?
Problem 4
The mean, median, and mode of the 7 data values are all equal to
. What is the value of
?
Problem 5
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, ). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?
Problem 6
A triangular array of 2016 coins in the first row, 2 coins in the second row, 3 coins in the third row, and so on up to coins in the
th row. What is the sum of the digits of
?
Problem 7
Which of these describes the graph of ?
Problem 8
What is the area of the shaded reigon of the given rectangle?
TODO: Diagram
Problem 9
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is , where
and
are positive integers. What is
?
TODO: Diagram
Problem 10
Five friends sat in a movie theater in a row containing seats, numbered
to
from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?
Problem 11
Each of the 100 students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?
Problem 12
Problem text
Problem 13
Problem text
Problem 14
Problem text
Problem 15
Problem text
Problem 16
Problem text
Problem 17
Problem text
Problem 18
Problem text
Problem 19
Problem text
Problem 20
Problem text
Problem 21
Problem text
Problem 22
Problem text
Problem 23
Problem text
Problem 24
Problem text
Problem 25
Problem text