2022年AMC12真题预测及答案.docx

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  bywhere  denotes the greatest integer less than or equal to . What is the value of ?

Solution
Problem 4
The mean, median, and mode of the  data values  are all equal 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?

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 , where  and  are positive integers. What is  ?

Solution
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?

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

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 side of the red ball. Observe that  and that approaches  as  grows large. 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  be the centers, respectively, of equilateral triangles with bases  and each exterior to the square. What is the ratio 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  and that  for all nonzero real numbers  and  (Here the dot  represents the usual multiplication operation.) The solution to the equation  can be written as  where  and  are relatively prime positive integers. What is 

Solution
Problem 21
A quadrilateral is inscribed in a circle of radius  Three of the sides of this quadrilateral have length  What is the length of its fourth side?

Solution
Problem 22
How many ordered triples  of positive integers satisfy  and ?

Solution
Problem 23
Three numbers in the interval  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  and , and thus . What is the sum of the digits of ?

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