Paper Set

2004

2004 AMC12 Paper & Solutions Pick Paper A

25 questions with solutions, combinatorics features recursive methods.

Sample Problems Download PDF

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25 Questions

75 Minutes

Max Score 150

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Exam Overview

This exam emphasizes combinatorics. Overall difficulty is moderate.

DDifficulty

EasyQ1-10
MediumQ11-20
HardQ21-25

TTopics

Algebra35%

Geometry25%

Number Theory15%

Combinatorics25%

AAwards

AIME Qualification

AIME Qualification (Top 2.5%)

89+

Honor Roll

Honor Roll (Top 5%)

79+

Achievement Roll

Grade 10 and below

64+

Sample Problems

2004 AMC12 Sample Problems (12 questions)

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

Q1EasyPolynomial

If f(x) = 2x² - 7x + 2, find f(2).

A) -8

B) -6

C) -2

D) 0

E) -4

Steps

1f(2) = 2×2² - 7×2 + 2

2= 2×4 - 14 + 2 = -4

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

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=3, d=6, find the 14-th term.

A) 81

B) 63

C) 72

D) 90

E) 99

Steps

1aₙ = a₁ + (n-1)d

2a_14 = 3 + 13×6 = 81

Answer: AGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 3ˣ = 27, find x.

A) 1

B) 3

C) 2

D) 4

E) 5

Steps

127 = 3^3

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 4?

A) 4

B) 6

C) 10

D) 8

E) 12

Steps

1⌊34 / 4⌋ = 8

Answer: DCount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 11-gon have?

A) 38

B) 41

C) 47

D) 50

E) 44

Steps

1diagonals = n(n-3)/2

2= 11×8/2 = 44

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 3^5 divided by 5.

A) 1

B) 2

C) 3

D) 4

E) 5

Steps

1compute 3^5 mod 5

2simplify step by step (modular arithmetic)

3=3

Answer: CModular arithmetic simplifies large exponents

Q22HardCompleting Sq

Find the minimum of f(x) = x² - 4x + 8.

A) 0

B) 2

C) 6

D) 4

E) 8

Steps

1complete the square:f(x) = (x - 2)² + 4

2when x = 2 , min at 4

Answer: DCompleting the square for quadratic extrema

Q24HardInclusion-Excl

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

A) 118

B) 134

C) 166

D) 182

E) 150

Steps

1total 3^5 = 243

2subtract empty boxes: -C(3,1)×2^5 = -96

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

4total 243 - 96 + 3 = 150

Answer: EInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

In △ABC, a=6, b=7, 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= 36 + 49 - 42 = 43

Answer: ALaw of cosines is key for solving triangles

Q1EasyPolynomial

If f(x) = 2x² - 9x + 2, find f(2).

A) -12

B) -10

C) -6

D) -4

E) -8

Steps

1f(2) = 2×2² - 9×2 + 2

2= 2×4 - 18 + 2 = -8

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

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=5, d=6, find the 14-th term.

A) 83

B) 65

C) 74

D) 92

E) 101

Steps

1aₙ = a₁ + (n-1)d

2a_14 = 5 + 13×6 = 83

Answer: AGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 3ˣ = 27, find x.

A) 1

B) 3

C) 2

D) 4

E) 5

Steps

127 = 3^3

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 4?

A) 5

B) 7

C) 11

D) 9

E) 13

Steps

1⌊36 / 4⌋ = 9

Answer: DCount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 13-gon have?

A) 59

B) 62

C) 68

D) 71

E) 65

Steps

1diagonals = n(n-3)/2

2= 13×10/2 = 65

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 3^5 divided by 5.

A) 1

B) 2

C) 3

D) 4

E) 5

Steps

1compute 3^5 mod 5

2simplify step by step (modular arithmetic)

3=3

Answer: CModular arithmetic simplifies large exponents

Q22HardCompleting Sq

Find the minimum of f(x) = x² - 4x + 10.

A) 2

B) 4

C) 8

D) 6

E) 10

Steps

1complete the square:f(x) = (x - 2)² + 6

2when x = 2 , min at 6

Answer: DCompleting the square for quadratic extrema

Q24HardInclusion-Excl

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

A) 118

B) 134

C) 166

D) 182

E) 150

Steps

1total 3^5 = 243

2subtract empty boxes: -C(3,1)×2^5 = -96

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

4total 243 - 96 + 3 = 150

Answer: EInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

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

A) 57

B) 45

C) 51

D) 63

E) 69

Steps

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

2= 64 + 49 - 56 = 57

Answer: ALaw of cosines is key for solving triangles

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2004 AMC12 Sample Problems (12 questions)

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