Competition · AMC preparation · step 4 of 4

AMC 10 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 10 2021B #1 grade 7+ arithmetic
How many integer values of x satisfy |x|<3π?
AMC 10 2021B #2 grade 8+ arithmetic
What is the value of √((3-2√3)²)+√((3+2√3)²)?
AMC 10 2021B #3 grade 6+ arithmetic
In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each cl…
AMC 10 2021B #4 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 10 2021B #5 grade 5+ arithmetic
The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied togeth…
AMC 10 2021B #6 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 10 2021B #7 grade 7+ geometry-2d
In a plane, four circles with radii 1,3,5, and 7 are tangent to line ell at the same point A, but they may be on either…
AMC 10 2021B #8 grade 5+ geometry-2d
Mr. Zhou places all the integers from 1 to 225 into a 15 by 15 grid. He places 1 in the middle square (eighth row and ei…
AMC 10 2021B #9 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 10 2021B #10 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 10 2021B #11 grade 6+ geometry-2d
Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equa…
AMC 10 2021B #12 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 10 2021B #13 grade 6+ arithmetic
Let n be a positive integer and d be a digit such that the value of the numeral 32d in base n equals 263, and the value…
AMC 10 2021B #14 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 10 2021B #15 grade 8+ algebra
The real number x satisfies the equation x+1/x = √5. What is the value of x¹¹-7x⁷+x³?
AMC 10 2021B #16 grade 4+ number-theory
Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, 1357,…
AMC 10 2021B #17 grade 2+ arithmetic
Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given 2 cards out of a set of 10 cards numbered 1,…
AMC 10 2021B #18 grade 7+ probability
A fair 6-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appe…
AMC 10 2021B #19 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 10 2021B #20 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 10 2021B #21 grade 8+ geometry-2d
A square piece of paper has side length 1 and vertices A,B,C, and D in that order. As shown in the figure, the paper is…
AMC 10 2021B #22 grade 7+ probability
Ang, Ben, and Jasmin each have 5 blocks, colored red, blue, yellow, white, and green; and there are 5 empty boxes. Each…
AMC 10 2021B #23 grade 8+ geometry-2d
A square with side length 8 is colored white except for 4 black isosceles right triangular regions with legs of length 2…
AMC 10 2021B #24 grade 7+ arithmetic
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 10 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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