Paper Set

2020

2020 AMC12 Paper & Solutions Pick Paper A

25 complete solutions, number theory weight increased with prime factorization and divisibility. Geometry focuses on area.

Sample Problems Download PDF

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25 Questions

75 Minutes

Max Score 150

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Exam Overview

>, Area.

DDifficulty

EasyQ1-10
MediumQ11-20
HardQ21-25

TTopics

Algebra30%

Geometry30%

Number Theory25%

Combinatorics15%

AAwards

AIME Qualification

AIME Qualification (Top 2.5%)

93+

Honor Roll

Honor Roll (Top 5%)

79+

Achievement Roll

Grade 10 and below

62+

Sample Problems

2020 AMC12 Sample Problems (12 questions)

Sample reference problems by difficulty — click an option to check your answer

Q1EasyPolynomial

If f(x) = 2x² - 3x + 3, find f(2).

A) 5

B) 1

C) 3

D) 7

E) 9

Steps

1f(2) = 2×2² - 3×2 + 3

2= 2×4 - 6 + 3 = 5

Answer: ASubstitute, then exponentiate, multiply/divide, add/subtract

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=3, d=2, find the 12-th term.

A) 19

B) 25

C) 22

D) 28

E) 31

Steps

1aₙ = a₁ + (n-1)d

2a_12 = 3 + 11×2 = 25

Answer: BGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 4ˣ = 64, find x.

A) 1

B) 2

C) 3

D) 4

E) 5

Steps

164 = 4^3

2so x = 3

Answer: CConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 5x + 4 = 0?

A) 1

B) 3

C) 7

D) 5

E) 9

Steps

1Vieta's:sum of roots = -(-5)/1 = 5

2product of roots = 4

Answer: DFor x²-px+q=0, sum of roots = p

Q10MediumDivisibility

How many integers from 1 to 30 are divisible by 5?

A) 2

B) 4

C) 8

D) 10

E) 6

Steps

1⌊30 / 5⌋ = 6

Answer: ECount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 9-gon have?

A) 27

B) 21

C) 24

D) 30

E) 33

Steps

1diagonals = n(n-3)/2

2= 9×6/2 = 27

Answer: An-gon diagonals: n(n-3)/2

Q15MediumCombination

Compute C(6, 3).

A) 14

B) 20

C) 17

D) 23

E) 26

Steps

1C(n,3) = n(n-1)(n-2)/6

2= 6×5×4/6 = 20

Answer: BCombination: C(n,k)=n!/(k!(n-k)!)

Q17MediumPythagorean

Right triangle legs 3 and 4, find the hypotenuse.

A) 1

B) 3

C) 5

D) 7

E) 9

Steps

1c² = 3² + 4² = 9 + 16 = 25

2c = √25 = 5

Answer: CPythagorean theorem a²+b²=c²

Q20HardModular Arith

Find the remainder of 3^6 divided by 5.

A) 2

B) 3

C) 5

D) 4

E) 6

Steps

1compute 3^6 mod 5

2simplify step by step (modular arithmetic)

3=4

Answer: DModular arithmetic simplifies large exponents

Q22HardCompleting Sq

Find the minimum of f(x) = x² - 4x + 9.

A) 1

B) 3

C) 7

D) 9

E) 5

Steps

1complete the square:f(x) = (x - 2)² + 5

2when x = 2 , min at 5

Answer: ECompleting the square for quadratic extrema

Q24HardInclusion-Excl

6 distinct balls into 3 distinct boxes, each ≥1 ball, how many ways?

A) 540

B) 430

C) 485

D) 595

E) 650

Steps

1total 3^6 = 729

2subtract empty boxes: -C(3,1)×2^6 = -192

3add back 2 empty boxes: +C(3,2)×1 = +3

4total 729 - 192 + 3 = 540

Answer: AInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

In △ABC, a=7, b=8, cos C=1/2, find c².

A) 45

B) 57

C) 51

D) 63

E) 69

Steps

1Law of cosines c² = a² + b² - 2ab·cos C

2= 49 + 64 - 56 = 57

Answer: BLaw of cosines is key for solving triangles

Q1EasyPolynomial

If f(x) = 2x² - 5x + 3, find f(2).

A) 1

B) 3

C) 5

D) 7

E) 9

Steps

1f(2) = 2×2² - 5×2 + 3

2= 2×4 - 10 + 3 = 1

Answer: ASubstitute, then exponentiate, multiply/divide, add/subtract

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=5, d=2, find the 12-th term.

A) 21

B) 27

C) 24

D) 30

E) 33

Steps

1aₙ = a₁ + (n-1)d

2a_12 = 5 + 11×2 = 27

Answer: BGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 4ˣ = 64, find x.

A) 1

B) 2

C) 3

D) 4

E) 5

Steps

164 = 4^3

2so x = 3

Answer: CConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 7x + 4 = 0?

A) 3

B) 5

C) 9

D) 7

E) 11

Steps

1Vieta's:sum of roots = -(-7)/1 = 7

2product of roots = 4

Answer: DFor x²-px+q=0, sum of roots = p

Q10MediumDivisibility

How many integers from 1 to 32 are divisible by 5?

A) 2

B) 4

C) 8

D) 10

E) 6

Steps

1⌊32 / 5⌋ = 6

Answer: ECount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 11-gon have?

A) 44

B) 38

C) 41

D) 47

E) 50

Steps

1diagonals = n(n-3)/2

2= 11×8/2 = 44

Answer: An-gon diagonals: n(n-3)/2

Q15MediumCombination

Compute C(8, 3).

A) 44

B) 56

C) 50

D) 62

E) 68

Steps

1C(n,3) = n(n-1)(n-2)/6

2= 8×7×6/6 = 56

Answer: BCombination: C(n,k)=n!/(k!(n-k)!)

Q17MediumPythagorean

Right triangle legs 6 and 8, find the hypotenuse.

A) 6

B) 8

C) 10

D) 12

E) 14

Steps

1c² = 6² + 8² = 36 + 64 = 100

2c = √100 = 10

Answer: CPythagorean theorem a²+b²=c²

Q20HardModular Arith

Find the remainder of 3^6 divided by 5.

A) 2

B) 3

C) 5

D) 4

E) 6

Steps

1compute 3^6 mod 5

2simplify step by step (modular arithmetic)

3=4

Answer: DModular arithmetic simplifies large exponents

Q22HardCompleting Sq

Find the minimum of f(x) = x² - 4x + 11.

A) 3

B) 5

C) 9

D) 11

E) 7

Steps

1complete the square:f(x) = (x - 2)² + 7

2when x = 2 , min at 7

Answer: ECompleting the square for quadratic extrema

Q24HardInclusion-Excl

6 distinct balls into 3 distinct boxes, each ≥1 ball, how many ways?

A) 540

B) 430

C) 485

D) 595

E) 650

Steps

1total 3^6 = 729

2subtract empty boxes: -C(3,1)×2^6 = -192

3add back 2 empty boxes: +C(3,2)×1 = +3

4total 729 - 192 + 3 = 540

Answer: AInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

In △ABC, a=9, b=8, cos C=1/2, find c².

A) 57

B) 73

C) 65

D) 81

E) 89

Steps

1Law of cosines c² = a² + b² - 2ab·cos C

2= 81 + 64 - 72 = 73

Answer: BLaw of cosines is key for solving triangles

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2020 AMC12 Paper & Solutions Pick Paper A

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2020 AMC12 Sample Problems (12 questions)

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