Yesterday, thousands of middle school and high school students participated in this year’s AMC 10B and 12B Competition. Hopefully everyone was able to take the exam safely, whether they took it online or in school!

The problems can now be discussed! See below for answer keys for both the Fall 2021-22 AMC 10B and AMC 12B questions as well as the concepts tested on each problem.

AMC 10B Answers

1

E

arithmetic and place values

2

B

geometry: areas

3

E

arithmetic: fractions, common denominators, and difference of two squares

4

C

algebra: equations and absolute values

5

E

arithmetic: exponents and division

6

B

number theory: prime factorizations and number of factors

7

C

algebra: fractions, completing the rectangle, and factors

8

C

number theory: primes and divisibility

9

C

algebra and arithmetic: ratios and fractions

10

A

logic and number theory: primes, perfect squares, and logical reasoning

11

B

geometry: areas, circles, regular hexagons, and equilateral triangles

12

D

algebra: equations, substitutions, and logical thinking

13

B

geometry: areas, triangles, and inscribed squares

14

C

combinatorics and number theory: probability, divisibility, and complementary counting

15

D

geometry: similar triangles and Pythagorean theorem

16

D

combinatorics: arrangements and expected values

17

D

geometry and algebra: lines and transformations

18

E

geometry: areas, regular polygons, and special angles

19

A

algebra: scientific notation, roots, and estimations

20

C

combinatorics: probability, symmetry, and conditional probability

21

E

geometry and combinatorics: polygons, angles, and patterns

22

B

number theory: modular arithmetic, sequences, and patterns

23

D

combinatorics: complementary counting, cases, and careful thinking

24

A

combinatorics: counting, symmetry, and spatial reasoning

25

E

geometry and algebra: similar triangles, lengths, and solving equations

AMC 12B Answers

1

E

arithmetic and place values

2

B

geometry: areas

3

C

algebra: equations and absolute values

4

E

arithmetic: exponents and division

5

C

algebra: fractions, completing the rectangle, and factors

6

C

number theory: primes and divisibility

7

D

algebra: equations, substitutions, and logical thinking

8

D

geometry: triangles, areas, and trigonometry

9

B

geometry: circles, triangles, and circumradii

10

E

algebra: unit triangle trigonometry and isosceles triangles

11

C

combinatorics and number theory: probability, divisibility, and complementary counting

12

B

number theory: prime factorization and sum of divisors

13

E

algebra: trigonometry and identities

14

B

algebra: complex numbers, polynomials, and careful thinking

15

E

geometry: areas, regular polygons, and special angles

16

B

number theory: cases, GCDs, and equations

17

A

combinatorics: probability, recursion, and states

18

A

combinatorics: sequences, recursion, and limits

19

E

geometry and combinatorics: polygons, angles, and patterns

20

A

combinatorics: counting, symmetry, and spatial reasoning

21

A

algebra: complex numbers, polar form, and substitutions

22

C

geometry: triangles, circles, tangent lines, and coordinates

23

A

combinatorics: counting and expected values

24

C

geometry: triangles, circles, Power of a Point, and Stewart’s theorem

25

C

number theory: remainders, factors, and cases

In total, 11 questions of the same questions appeared on both the AMC 10B and AMC 12B. They are listed below:

Interested in seeing the full problem sets? The 2021-22 AMC 10B and 12B problems will also be available on the ZIML site along with previous exams. Click here for AMC 10 and here for AMC 12.

The AMC 10 and 12 exams are administered by the Mathematical Association of America (MAA). For general information about the AMC 10 and 12, click here.