2016年AMC12真题及答案

2016 AMC12 A

Problem 1

What is the value of ?

Solution Problem 2

For what value of does

?

Solution Problem 3

The remainder can be defined for all real numbers and with

by

where

What is the value of

?

denotes the greatest integer less than or equal to .

Solution Problem 4

The mean, median, and mode of the data values to . What is the value of ?

Solution 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?

are all equal

Solution Problem 6

A triangular array of coins has coin in the first row, coins in the second row, coins in the third row, and so on up to coins in the th row. What is the sum of the digits of ?

Solution Problem 7

Which of these describes the graph of

?

Solution Problem 8

What is the area of the shaded region of the given

rectangle?

Solution 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 integers. What is

?

, where and are positive

Solution Problem 10

Five friends sat in a movie theater in a row containing seats, numbered to from left to right. (The directions \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?

Solution Problem 11

Each of the 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 students who cannot sing, students who cannot dance, and students who cannot act. How many students have two of these talents?

Solution Problem 12

In , , , and and bisects . Point lies on intersect at . What is the ratio : ?

. Point

, and lies on bisects

,

. The bisectors

Solution Problem 13 Let

be a positive multiple of . One red ball and

green balls are arranged in a line in

random order. Let be the probability that at least of the green balls are on the same

and that

approaches as

?

grows large.

side of the red ball. Observe that

What is the sum of the digits of the least value of such that

Solution Problem 14

Each vertex of a cube is to be labeled with an integer from through , with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?

Solution Problem 15

Circles with centers and , having radii and , respectively, lie on the same side of line and are tangent to at and , respectively, with between and . The

circle with center is externally tangent to each of the other two circles. What is the area of triangle ?

Solution Problem 16

The graphs of and are plotted on the

same set of axes. How many points in the plane with positive -coordinates lie on two or more of the graphs?

Solution Problem 17 Let

be a square. Let

and and

be the centers, respectively, of equilateral each exterior to the square. What is the ratio

?

triangles with bases of the area of square

to the area of square

Solution Problem 18

For some positive integer the number has positive integer divisors, including and the number How many positive integer divisors does the number have?

Solution Problem 19

Jerry starts at on the real number line. He tosses a fair coin times. When he gets heads, he moves unit in the positive direction; when he gets tails, he moves unit in the negative direction. The probability that he reaches at some time during this process is

where and are relatively prime positive integers. What is

)

(For example, he

succeeds if his sequence of tosses is

Solution Problem 20

A binary operation has the properties that that

for all nonzero real numbers

and

and (Here the dot represents the usual

can be

multiplication operation.) The solution to the equation written as

where and are relatively prime positive integers. What is

Solution Problem 21

A quadrilateral is inscribed in a circle of radius have length

What is the length of its fourth side?

Solution Problem 22

How many ordered triples satisfy

of positive integers

and

Solution Problem 23

Three numbers in the interval

?

Three of the sides of this quadrilateral

are chosen independently and at random. What is the

probability that the chosen numbers are the side lengths of a triangle with positive area?

Solution Problem 24

There is a smallest positive real number such that there exists a positive real number such that all the roots of the polynomial are real. In fact, for this value of the value of is unique. What is the value of

Solution Problem 25

Let be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with digits. Every time Bernardo writes a number, Silvia erases the last digits of it. Bernardo then writes the next perfect square, Silvia erases the last digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let

be the smallest

positive integer not written on the board. For example, if , then the numbers that Bernardo writes are , and the numbers showing on the board after Silvia erases are of

and , and thus

?

. What is the sum of the digits

2016 AMC 12A Answer Key

1 B 2 C 3 B 4 D 5 E 6 D 7 D 8 D 9 E 10 B 11 E 12 C 13 A 14 C 15 D 16 D 17 B 18 D 19 B 20 A 21 E 22 A 23 C 24 B 25 E

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