Paper Set

2024

2024 AMC12 Paper & Solutions Pick Paper A

25 complete solutions with difficulty tags and topic distribution. Features: 3D unfolding, circular arrangements.

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25 Questions

75 Minutes

Max Score 150

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Exam Overview

This exam emphasizes multiple math topics. Overall difficulty is moderate.

DDifficulty

EasyQ1-10
MediumQ11-20
HardQ21-25

TTopics

Algebra35%

Geometry35%

Number Theory15%

Combinatorics15%

AAwards

AIME Qualification

AIME Qualification (Top 2.5%)

85+

Honor Roll

Honor Roll (Top 5%)

75+

Achievement Roll

Grade 10 and below

60+

Sample Problems

2024 AMC12 Sample Problems (12 questions)

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

Q1EasyPolynomial

If f(x) = 2x² - 7x + 1, find f(2).

A) -9

B) -7

C) -3

D) -1

E) -5

Steps

1f(2) = 2×2² - 7×2 + 1

2= 2×4 - 14 + 1 = -5

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

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=3, d=6, find the 10-th term.

A) 57

B) 45

C) 51

D) 63

E) 69

Steps

1aₙ = a₁ + (n-1)d

2a_10 = 3 + 9×6 = 57

Answer: AGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 2ˣ = 8, find x.

A) 1

B) 3

C) 2

D) 4

E) 5

Steps

18 = 2^3

2so x = 3

Answer: BConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 9x + 4 = 0?

A) 5

B) 7

C) 9

D) 11

E) 13

Steps

1Vieta's:sum of roots = -(-9)/1 = 9

2product of roots = 4

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

Q10MediumDivisibility

How many integers from 1 to 34 are divisible by 3?

A) 7

B) 9

C) 13

D) 11

E) 15

Steps

1⌊34 / 3⌋ = 11

Answer: DCount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 7-gon have?

A) 8

B) 11

C) 17

D) 20

E) 14

Steps

1diagonals = n(n-3)/2

2= 7×4/2 = 14

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

Q15MediumCombination

Compute C(10, 3).

A) 120

B) 94

C) 107

D) 133

E) 146

Steps

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

2= 10×9×8/6 = 120

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

Q17MediumPythagorean

Right triangle legs 7 and 24, find the hypotenuse.

A) 21

B) 25

C) 23

D) 27

E) 29

Steps

1c² = 7² + 24² = 49 + 576 = 625

2c = √625 = 25

Answer: BPythagorean theorem a²+b²=c²

Q20HardModular Arith

Find the remainder of 3^4 divided by 5.

A) 0

B) 2

C) 1

D) 3

E) 4

Steps

1compute 3^4 mod 5

2simplify step by step (modular arithmetic)

3=1

Answer: CModular arithmetic simplifies large exponents

Q22HardCompleting Sq

Find the minimum of f(x) = x² - 4x + 7.

A) 1

B) 5

C) 7

D) 3

E) 9

Steps

1complete the square:f(x) = (x - 2)² + 3

2when x = 2 , min at 3

Answer: DCompleting the square for quadratic extrema

Q24HardInclusion-Excl

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

A) 26

B) 31

C) 41

D) 46

E) 36

Steps

1total 3^4 = 81

2subtract empty boxes: -C(3,1)×2^4 = -48

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

4total 81 - 48 + 3 = 36

Answer: EInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

In △ABC, a=5, b=6, cos C=1/2, find c².

A) 31

B) 23

C) 27

D) 35

E) 39

Steps

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

2= 25 + 36 - 30 = 31

Answer: ALaw of cosines is key for solving triangles

Q1EasyPolynomial

If f(x) = 2x² - 9x + 1, find f(2).

A) -13

B) -11

C) -7

D) -5

E) -9

Steps

1f(2) = 2×2² - 9×2 + 1

2= 2×4 - 18 + 1 = -9

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

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=5, d=6, find the 10-th term.

A) 59

B) 47

C) 53

D) 65

E) 71

Steps

1aₙ = a₁ + (n-1)d

2a_10 = 5 + 9×6 = 59

Answer: AGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 2ˣ = 8, find x.

A) 1

B) 3

C) 2

D) 4

E) 5

Steps

18 = 2^3

2so x = 3

Answer: BConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 11x + 4 = 0?

A) 7

B) 9

C) 11

D) 13

E) 15

Steps

1Vieta's:sum of roots = -(-11)/1 = 11

2product of roots = 4

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

Q10MediumDivisibility

How many integers from 1 to 36 are divisible by 3?

A) 8

B) 10

C) 14

D) 12

E) 16

Steps

1⌊36 / 3⌋ = 12

Answer: DCount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 9-gon have?

A) 21

B) 24

C) 30

D) 33

E) 27

Steps

1diagonals = n(n-3)/2

2= 9×6/2 = 27

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

Q15MediumCombination

Compute C(12, 3).

A) 220

B) 174

C) 197

D) 243

E) 266

Steps

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

2= 12×11×10/6 = 220

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

Q17MediumPythagorean

Right triangle legs 5 and 12, find the hypotenuse.

A) 9

B) 13

C) 11

D) 15

E) 17

Steps

1c² = 5² + 12² = 25 + 144 = 169

2c = √169 = 13

Answer: BPythagorean theorem a²+b²=c²

Q20HardModular Arith

Find the remainder of 3^4 divided by 5.

A) 0

B) 2

C) 1

D) 3

E) 4

Steps

1compute 3^4 mod 5

2simplify step by step (modular arithmetic)

3=1

Answer: CModular arithmetic simplifies large exponents

Q22HardCompleting Sq

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

A) 1

B) 3

C) 7

D) 5

E) 9

Steps

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

2when x = 2 , min at 5

Answer: DCompleting the square for quadratic extrema

Q24HardInclusion-Excl

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

A) 26

B) 31

C) 41

D) 46

E) 36

Steps

1total 3^4 = 81

2subtract empty boxes: -C(3,1)×2^4 = -48

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

4total 81 - 48 + 3 = 36

Answer: EInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

In △ABC, a=7, b=6, cos C=1/2, find c².

A) 43

B) 33

C) 38

D) 48

E) 53

Steps

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

2= 49 + 36 - 42 = 43

Answer: ALaw of cosines is key for solving triangles

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2024 AMC12 Sample Problems (12 questions)

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