Competition · AMC preparation · step 4 of 4

AMC 12 2021B: all 25 problems

Each problem has a solution worked from first principles and the first grade (by CCSS standards) that can solve it.

AMC 12 2021B #1 grade 7+ arithmetic
How many integer values of x satisfy |x|<3π?
AMC 12 2021B #2 grade 4+ counting
At a math contest, 57 students are wearing blue shirts, and another 75 students are wearing yellow shirts. The 132 stude…
AMC 12 2021B #3 grade 7+ algebra
Suppose2+1/(1+1/(2+2/(3+x)))=144/53.What is the value of x?
AMC 12 2021B #4 grade 6+ rate-ratio
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is 84, and the a…
AMC 12 2021B #5 grade 8+ geometry-2d
The point P(a,b) in the xy-plane is first rotated counterclockwise by 90° around the point (1,5) and then reflected abou…
AMC 12 2021B #6 grade 8+ geometry-3d
An inverted cone with base radius 12 cm and height 18 cm is full of water. The water is poured into a tall cylinder whos…
AMC 12 2021B #7 grade 6+ number-theory
Let N = 34 · 34 · 63 · 270. What is the ratio of the sum of the odd divisors of N to the sum of the even divisors of N?
AMC 12 2021B #8 grade 8+ geometry-2d
Three equally spaced parallel lines intersect a circle, creating three chords of lengths 38,38, and 34. What is the dist…
AMC 12 2021B #9 grade 11+ algebra
AMC 12 2021B #10 grade 9+ number-theory
Two distinct numbers are selected from the set {1,2,3,4,…,36,37} so that the sum of the remaining 35 numbers is the prod…
AMC 12 2021B #11 grade 8+ geometry-2d
Triangle ABC has AB=13,BC=14 and AC=15. Let P be the point on AC such that PC=10. There are exactly two points D and E o…
AMC 12 2021B #12 grade 6+ algebra
Suppose that S is a finite set of positive integers. If the greatest integer in S is removed from S, then the average va…
AMC 12 2021B #13 grade 11+ algebra
How many values of θ in the interval 0<θ≤ 2π satisfy 1-3sinθ+5cos3θ = 0?
AMC 12 2021B #14 grade 10+ geometry-3d
Let ABCD be a rectangle and let DM be a segment perpendicular to the plane of ABCD. Suppose that DM has integer length,…
AMC 12 2021B #15 grade 8+ geometry-2d
The figure is constructed from 11 line segments, each of which has length 2. The area of pentagon ABCDE can be written a…
AMC 12 2021B #16 grade 11+ algebra
Let g(x) be a polynomial with leading coefficient 1, whose three roots are the reciprocals of the three roots of f(x)=x³…
AMC 12 2021B #17 grade 9+ geometry-2d
Let ABCD be an isosceles trapezoid having parallel bases AB and CD with AB>CD. Line segments from a point inside ABCD to…
AMC 12 2021B #18 grade 11+ algebra
Let z be a complex number satisfying 12|z|²=2|z+2|²+|z²+1|²+31. What is the value of z+6/z?
AMC 12 2021B #19 grade 8+ probability
Two fair dice, each with at least 6 faces are rolled. On each face of each die is printed a distinct integer from 1 to t…
AMC 12 2021B #20 grade 11+ algebra
Let Q(z) and R(z) be the unique polynomials such thatz²⁰²¹+1=(z²+z+1)Q(z)+R(z)and the degree of R is less than 2. What i…
AMC 12 2021B #21 grade 11+ algebra
Let S be the sum of all positive real numbers x for whichx^(2^√2)=√2^(2^x).Which of the following statements is true?
AMC 12 2021B #22 grade 7+ logic
Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a se…
AMC 12 2021B #23 grade 11+ probability
Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, t…
AMC 12 2021B #24 grade 11+ geometry-2d
Let ABCD be a parallelogram with area 15. Points P and Q are the projections of A and C, respectively, onto the line BD;…
AMC 12 2021B #25 grade 8+ number-theory
Let S be the set of lattice points in the coordinate plane, both of whose coordinates are integers between 1 and 30, inc…

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