Yesterday, thousands of middle school and high school students participated in this year’s AMC 10A and 12A 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 10A and AMC 12A questions as well as the concepts tested on each problem.

AMC 10A Answers

1

C

arithmetic and factors

2

E

geometry: areas, rectangles, and logic

3

D

geometry: volumes of cubes and spheres and approximations

4

B

algebra: rates and average speeds

5

E

number theory: primes, digits, and divisibility

6

B

algebra: rates, ratios, and careful thinking

7

D

geometry: angles in triangles and squares

8

B

number theory and algebra: digits and sums

9

E

combinatorics: probability

10

B

algebra: averages and weighted averages

11

A

algebra: rates and relative motion

12

D

number theory: number bases and modular arithmetic

13

D

combinatorics: probability and combinations

14

D

algebra: graphing, absolute values, parabolas

15

C

geometry: circles, tangent lines, and circumcircles

16

D

algebra: absolute values, floor functions, and symmetry

17

D

geometry: regular hexagons, planes, and similarity

18

C

combinatorics: cases, careful thinking, and symmetry

19

A

geometry: areas, circular regions, and careful thinking

20

B

algebra: quadratics and discriminants

21

E

combinatorics: arrangements and probability

22

B

geometry: solid geometry, circles, and equilateral triangles

23

D

number theory: prime factorizations, number of factors, and patterns

24

E

combinatorics: cases and careful thinking

25

A

algebra: quadratics, polynomials, and roots

AMC 12A Answers

1

C

arithmetic and factors

2

E

geometry: areas, rectangles, and logic

3

B

algebra: rates and average speeds

4

E

number theory: primes, digits, and divisibility

5

B

algebra: rates, ratios, and careful thinking

6

D

geometry: angles in triangles and squares

7

B

algebra: averages and weighted averages

8

D

number theory: prime factorizations and least common multiples

9

E

algebra and geometry: surface area, prisms, and logarithms

10

D

number theory: number bases and modular arithmetic

11

E

geometry: circles, chords, and power of a point

12

C

combinatorics and algebra: binomial expansion and binomial theorem

13

A

geometry and algebra: slopes, lines, and angle bisector theorem

14

E

geometry: areas, congruent triangles, and Law of Cosines

15

D

algebra: polynomials, complex roots, and Vieta’s theorem

16

B

combinatorics: pigeonhole principle and worst-case scenarios

17

B

algebra: quadratics and discriminants

18

E

combinatorics: arrangements and probability

19

B

geometry: trigonometry, identities, and quadratics

20

D

number theory: prime factorizations, number of factors, and patterns

21

C

geometry: similar triangles, trapezoids, and algebra

22

D

combinatorics: cases and complementary counting

23

A

algebra: quadratics, polynomials, and roots

24

E

geometry and algebra: quadrilaterals, arithmetic sequences, and equations

25

E

number theory and combinatorics: modular arithmetic, residue classes, inclusion-exclusion

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

AMC 10A

AMC 12A

1

1

2

2

4

3

5

4

6

5

7

6

10

7

12

10

20

17

21

18

23

20

25

23

Want full solutions to all the 2021 AMC 10A & 12A questions? Our full solutions review video has you covered! Registration and solutions will be available on Friday, November 12th. Click here for registration. The 2021 AMC 10A and 12A 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.