# 官方解禁首发！AMC10/12A考卷新鲜出炉，翰林真题解析大放送！

AMC10/12 A卷完美收官！

最新考试真题速来看！

01

AMC10 A卷考试真题

1.What is the value of (22−2)-(32−3)+(42−4)?

(A) 1  (B) 2  (C) 5  (D) 8  (E) 12

2. Portia’s high school has 3 times as many students as Lara’s high school. The two high  schools have a total of 2600 students. How many students does Portia’s high school  have?

(A) 600  (B) 650  (C) 1950  (D) 2000  (E) 2050

3.The sum of two natural numbers is 17,402. One of the two numbers is divisible by  10. If the units digit of that number is erased, the other number is obtained. What is the  difference of these two numbers?

(A) 10,272  (B) 11,700  (C) 13,362  (D) 14,238  (E) 15,426

4.A cart rolls down a hill, traveling 5 inches the first second and accelerating so that  during each successive 1-second time interval, it travels 7 inches more than during the  previous 1-second interval. The cart takes 30 seconds to reach the bottom of the hill.  How far, in inches, does it travel?

(A) 215  (B) 360  (C) 2992  (D) 3195  (E) 3242

5.The quiz scores of a class with k>12 students have a mean of 8. The mean of a  collection of 12 of these quiz scores is 14. What is the mean of the remaining quiz  scores in terms of k?

6.Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a  heavy backpack and walks slower. Chantal starts walking at 4 miles per hour. Halfway  to the tower, the trail becomes really steep, and Chantal slows down to 2 miles per hour.  After reaching the tower, she immediately turns around and descends the steep part of  the trail at 3 miles per hour. She meets Jean at the halfway point. What was Jean’s  average speed, in miles per hour, until they meet?

7. Tom has a collection of 13 snakes, 4 of which are purple and 5 of which are happy.  He observes that

 all of his happy snakes can add

 none of his purple snakes can subtract, and

 all of his snakes that can’t subtract also can’t add.

Which of these conclusions can be drawn about Tom’s snakes?

(B) Purple snakes are happy.

(C) Snakes that can add are purple.

(D) Happy snakes are not purple.

(E) Happy snakes can’t subtract.

8.When a student multiplied the number 66 by the repeating decimal,

Where a and b are digits, he did not notice the notation and just multiplied 66 times

Later he found that his answer is 0.5 less than the correct answer. What is the 2- digit integer ab ?

(A) 15  (B) 30  (C) 45  (D) 60  (E) 75

9.What is the least possible value of

for real numbers x and y?

10.Which of the following is equivalent to

11.For which of the following integers b is the base-b number 2021b−221b not divisible  by 3?

(A) 3  (B) 4  (C) 6  (D) 7  (E) 8

12.Two right circular cones with vertices facing down as shown in the figure below  contain the same amount of liquid. The radii of the top of the liquid surfaces are 3cm  and 6cm. Into each cone is dropped a spherical marble of radius 1cm, which sinks to  the bottom and is completely submerged without spilling any liquid. What is the ratio  of the rise of the liquid level in the narrow cone to the liquid level in the wide cone?

(A) 1:1  (B) 47:43  (C) 2:1  (D) 40:13  (E) 4:1

13.What is the volume of tetrahedron ABCD with edge lengths AB=2, AC=3, AD=4

and CD=5?

14.Ali the roots of polynomial

are positive  integers, possibly repeated. What is the value of B?

(A) -88  (B) -80  (C) -64  (D) -41  (E) -40

15.Values for A, B, C, and D are to be selected from {1, 2, 3, 4, 5, 6} without  replacement (i.e., no two letters have the same value). How many ways are there to  make such choices so that the two curves y = Ax2+B and y = Cx2+D intersect? (The order in which the curves are listed does not matter; for example, the choices A=3.  B=2, C=4, D=1 is considered the same as the choices A = 4, B=1, C= 3. D=2.)

16.In the following list of numbers, the integer n appears n times in the list for 1≤n ≤200

What is the median of the numbers in this list?

(A) 100.5  (B) 134  (C) 142  (D) 150.5  (E) 167

17.

(A) 65  (B) 132  (C) 157  (D) 194  (E) 215

18.

19.The area of the region bounded by the graph of

is m+nπ, where m and n are integers. What is m+n?

(A) 18  (B) 27  (C) 36  (D) 45  (E) 54

20.In how many ways can the sequence 1,2,3,4,5 be rearranged so that no three  consecutive terms are increasing and no three consecutive terms are decreasing?

(A) 10  (B) 18  (C) 24  (D) 32  (E) 44

21.

(A) 47  (B) 52  (C) 55  (D) 58  (E) 63

22.Hiram's algebra notes are 50 pages long and are printed on 25 sheets of paper: the  first sheet contains pages 1 and 2 the second sheet contains pages 3 and 4. and so on.  One day he leaves his notes on the table before leaving for lunch. and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back,  he discovers that his roommate has taken a consecutive set of sheets from the notes and  that the average (mean) of the page numbers on all remaining sheets is exactly 19 How  many sheets were borrowed?

(A) 10  (B) 13  (C) 15  (D) 17  (E) 20

23.Frieda the frog begins a sequence of hops on a 3×3 grid of squares, moving one  square on each hop and choosing at random the direction of each hop up, down, left, or  right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge. For example If Frieda  begins in the center square and makes two hops "up”, the first hop would place her In  the top row middle square, and the second hop would cause Frieda to Jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the  center square, makes at most four hops at random, and stops hopping If she lands on a  comer square. What Is the probability that she reaches a corner square on one of the  four hops?

24.

25.How many ways are there to place 3 Indistinguishable red chips, 3 indistinguishable  blue chips, and 3 Indistinguishable green chips in the squares of a 3×3 grid so that no  two chips of the same color are directly adjacent 10 each other, either vertically or horizontally？

(A) 12  (B) 18  (C) 24  (D) 30  (E) 36

02

AMC12 A卷考试真题

1.What is the value of

(A) 0  (B) 50  (C) 52  (D) 54  (E) 57

2.Under what conditions is

true, where a and b are real numbers?

(A) It is never true

(B) It is true if and only if ab=0

(C) It is true if and only if a+b≥0

(D) It is true if and only if ab=0 and a+b≥0

(E) It is always true

3.The sum of two natural numbers is 17,402. One of the two numbers is divisible by  10. If the units digit of that number is erased, the other number is obtained. What is the  difference of these two numbers?

(A) 10,272(B) 11,700 (C) 13,,362  (D) 14,238  (E) 15,426

4.Tom has a collection of 13 snakes, 4 of which are purple and 5 of which are happy.  He observes that

 all of his happy snakes can add

 none of his purple snakes can subtract, and

 all of his snakes that can’t subtract also can’t add

Which of these conclusions can be drawn about Tom’s snakes?

(B) Purple snakes are happy.

(C) Snakes that can add are purple.

(D) Happy snakes are not purple.

(E) Happy snakes can’t subtract.

5.When a student multiplied the number 66 by the repeating decimal,

Where a and b are digits, he did not notice the notation and just multiplied 66 times 1.ab . Later he found that his answer is 0.5 less than the correct answer. What is the 2- digit integer ab ?

(A) 15  (B) 30  (C) 45  (D) 60  (E) 75

6.A deck of cards has only red cards and black cards. The probability of a randomly  chosen card being red is  1/3. When 4 black cards are added to the deck, the probability  of choosing red becomes 1/4 . How many cards were in the deck originally?

(A) 6  (B) 9  (C) 12  (D) 15  (E) 18

7.What is the least possible value of

for real numbers x and y?

8.A sequence of numbers is defined by D0 = 0, D1 = 0, D2 =1 and Dn= Dn-1 + Dn-3 for  n≥3. What are the parities (evenness or oddness) of the triple of numbers (D2021, D2022,  D2013), where E denotes even and 0 denotes odd

(A) (O, E, O)

(B) (E, E, O)

(C) (E, O, E)

(D) (O, O, E)

(E) (O, O, O)

9.Which of the following is equivalent to

10.Two right circular cones with vertices facing down as shown in the figure below  contain the same amount of liquid. The radii of the top of the liquid surfaces are 3cm  and 6cm. Into each cone is dropped a spherical marble of radius 1cm, which sinks to  the bottom and is completely submerged without spilling any liquid. What is the ratio  of the rise of the liquid level in the narrow cone to the liquid level in the wide cone?

(A) 1:1  (B) 47:43  (C) 2:1  (D) 40:13  (E) 4:1

11.A laser is placed at the point (3, 5). The laser beam travels in a straight line. Larry  wants the beam to hit and bounce off the y-axis, then hit and bounce off the x-axis, the  hit the point (7, 5). What is the total distance the beam will travel along this path?

12.Ali the roots of polynomial are positive

integers, possibly repeated. What is the value of B?

(A) -88  (B) -80  (C) -64  (D) -41  (E) -40

13.Of the following complex number z, which one has the property that z5 has the  greatest real part?

14.

15.A choir director must select a group of singers from among his 6 tenors and 8 basses.  The only requirements are that the difference between the numbers of tenors and basses  must be a multiple of 4, and the group must have at least one singer. Let N be the number  of groups that could be selected. What is the remainder when N is divided by 100?

(A) 47  (B) 48  (C) 83  (D) 95  (E) 9616.In the following list of numbers, the integer n appears n times in the list for 1≤n ≤200

What is the median of the numbers in this list?

(A) 100.5  (B) 134  (C) 142  (D) 150.5  (E) 167

17.

(A) 65  (B) 132  (C) 157  (D) 194  (E) 215

18.

19.How many solutions does the equation sin

have in the  closed interval [0,π ]

(A) 0  (B) 1  (C) 2  (D) 3  (E) 4

20.Suppose that on a parabola with vertex V and focus F there exists a point A such  that AF=20 and AV=21. What is the sum of all possible values of the length FV?

21.

(A) 7  (B) 9  (C) 11  (D) 13  (E) 15

22.

23.Frieda the frog begins a sequence of hops on a 3×3 grid of squares, moving one  square on each hop and choosing at random the direction of each hop up, down, left, or  right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge. For example If Frieda  begins in the center square and makes two hops "up”, the first hop would place her In  the top row middle square, and the second hop would cause Frieda to Jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the  center square, makes at most four hops at random, and stops hopping If she lands on a  comer square. What Is the probability that she reaches a corner square on one of the  four hops?

24.

(A) 110  (B) 114  (C) 118  (D) 122  (E) 126

25.Let d(n) denote the number of positive integers that divide n, including 1 and n. For  example, d(1)=1, d(2)=2, and d(12)=6. (This function is known as the divisor function.)  Let

There is a unique positive integer N such that f(N)>f(n) for all positive integers n≠N. what is the sum of the digits of N?

(A) 5  (B) 6  (C) 7  (D) 8  (E) 9

AMC10/12 B卷将2月11日准时开考

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