2021年11月AMC12 B卷真题及解析

2021 AMC 12B

1. What is the value of 1234 + 2341 + 3412 + 4123?

(A) 10,000  (B) 10,010  (C) 10,110  (D) 11,000  (E) 11,110

 

2. What is the area of the shaded figure shown below?

(A) 4     (B) 6     (C) 8     (D) 10    (E) 12

 

3. At noon on a certain day, Minneapolis is N degrees warmer than St. Louis.At 4:00 the temperature in Minneapolis has fallen by 5 degrees while the temperature in Louis has risen by 3 degrees, at which time the temperatures in the two cities differ by 2 degrees. What is the product of all possible values of N?

(A) 10    (B) 30    (C) 60    (D) 100   (E) 120

 

4 Let n=8²⁰²² . Which of the following is equal to

(A)  4¹⁰¹⁰  (B)  2²⁰²² (C)  8²⁰¹⁸ (D)4³⁰³¹    (E)4³⁰³²

 

5.  Call a fraction   not necessarily in simplest form, special if a and b are positive integers whose sum is 15. How many distinct integers can be writtenas the sum of two, not necessarily different, special fractions?

(A) 9     (B) 10    (C) 11    (D) 12    (E) 13

 

6. The greatest prime number that is a divisor of 16,384 is 2 because 16,384 =2¹⁴. What is the sum of the digits of the greatest prime number that is adivisor of 16,383?

(A) 3     (B) 7     (C) 10    (D) 16    (E) 22

 

7. Which of the following conditions is sufficient to guarantee that integersx, y, and z satisfy the equation

x(x-y)+y(y-z)+z(z-x)=1?

(A) x>y and y=z  (B) x=y1 and y=z1  (C) x=z+1 and y=x+1 (D) x=z and y1=x  (E) x+y+z=1

 

8. The product of the lengths of the two congruent sides of an obtuse isoscelestriangle is equal to the product of the base and twice the triangle's heightto the base. What is the measure, in degrees, of the vertex angle of this triangle?

(A) 105   (B) 120   (C) 135   (D) 150   (E) 165

 

9. Triangle ABCis equilateral with side length 6. Suppose that O is the centerof the inscribed circle of this triangle. What is the area of the circle passing through A, O and C?

(A)9π    (B)12π    (C)18π    (D)24π    (E)27π

 

10. What is the sum of all possible values of t between 0 and 360 such thatthe triangle in the coordinate plane whose vertices are (cos 40°, sin 40°), (cos 60°, sin 60°), and (cos t^o, sin t^o) is isosceles?

(A) 100   (B) 150   (C) 330   (D) 360   (E) 380

 

11. Una rolls 6 standard 6-sided dice simultaneously and calculates the productof the 6 numbers obtained. What is the probability that the product isdivisible by 4?

 

12. For n a positive integer, let f(n) be the quotient obtained when the sum of all positive divisors of n is divided by n. For example,

What is f（768）- f（384）?

 

13.  Let  What is the value of

 

14.  Suppose that P(z), Q(z), and R(z) are polynomials with real coefficients, having degrees 2, 3, and 6, respectively, and constant terms 1, 2, and 3,respectively. Let N be the number of distinct complex numbers z that satisfythe equation P(z). Q(z) = R(z). What is the minimum possible value of N?

(A) 0     (B) 1     (C) 2     (D) 3    (E) 5

15. Three identical square sheets of paper each with side length 6 are stacked on top of each other. The middle sheet is rotated clockwise 30°  about its center and the top sheet is rotated clockwise 60° about its center. Resulting in the 24-sided polygon shown in the figure below. The area of this polygon can be expressed in the form   where a, b, and c are positive integers, and c is not divisible by the square of any prime. What is a+b+c?

(A) 75    (B) 93    (C) 96    (D) 129   (E) 147

 

16. Suppose a, b, and c are positive integers such that a + b + c = 23 andgcd(a, b) + gcd(b, c) + gcd(c, a) = 9. What is the sum of all possible distinctvalues of a² +b²+ c²?

(A) 259   (B) 438   (C) 516   (D) 625   (E) 687

 

17. A bug starts at a vertex of a grid made up of equilateral triangles of sidelength l. At each step the bug moves in one of the 6 possible directionsalong the grid lines randomly and independently with equal probability. What is the probability that after 5 moves the bug never will have been more than 1 unit away from the starting position?

18. Set and for k≥ 0 let  be determined by the recurrence . This sequence tends to a limit; call it L. What is the least value of k such that

(A) 10    (B) 97    (C) 123   (D) 329   (E) 401

 

19. Regular polygons with 5,6,7, and 8 sides are inscribed in the same circle.No two of the polygons share a vertex, and no three of their sides intersectat a common point. At how many points inside the circle do two of their sides intersect?

(A) 52    (B) 56    (C) 60    (D) 64    (E) 68

 

20. A cube is constructed from 4 white unit cubes and 4 blue unit cubes. How many different ways are there to construct the  2X2X2 cube using thesesmaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)

(A) 7     (B) 8     (C) 9     (D) 10    (E) 11

 

21. 2 For real numbers x, let

(A) 0     (B) 1     (C) 2     (D) 3    (E) 4

 

22. Right triangle ABC has side lengths BC = 6, AC = 8, and AB = 10. A circle centered at O is tangent to line BC at B and passes through A. A circle centered at P is tangent to line AC at A and passes through B. What is OP?

 

23. What is the average number of pairs of consecutive integers in a randomly selected subset of 5 distinct integers chosen from the set {1,2,3,…,30}? (For example, the set {1,17,18,19,30} has 2 pairs of consecutive integers.)

 

24. Triangle ABC has side lengths AB=11. BC=24, and CA=20. The bisector of ∠BAC intersects in point D and intersects the circumcircle of △ABC  in point  E≠A. The circumcircle of △BED intersects the line AB in points B F≠B. What is CF?

(A) 28    (B)    (C) 30    (D) 32   (E)

 

25. For n a positive integer, let R(n) be the sum of remainders when n is divided by 2, 3, 4, 5, 6, 7, 8, 9, and 10. For example, R(15) = 1 + 0 + 3 + 0 + 3 + 1 + 7 + 6 + 5 = 26. How many two-digit positive integers n satisfy R(n) = R(n + 1)?

(A) 0     (B) 1     (C) 2     (D) 3     (E) 4

 

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