Paper Set

2008

2008 AMC12 Paper & Solutions Pick Paper A

25 questions with solutions, algebra focuses on functions and graphs, challenging combinatorics.

Sample Problems Download PDF

Scan to get free 2008 PDF + solutions

25 Questions

75 Minutes

Max Score 150

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Exam Overview

This exam emphasizes algebra, combinatorics. Overall difficulty is moderate.

DDifficulty

EasyQ1-10
MediumQ11-20
HardQ21-25

TTopics

Algebra35%

Geometry25%

Number Theory20%

Combinatorics20%

AAwards

AIME Qualification

AIME Qualification (Top 2.5%)

93+

Honor Roll

Honor Roll (Top 5%)

75+

Achievement Roll

Grade 10 and below

62+

Sample Problems

2008 AMC12 Sample Problems (12 questions)

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

Q1EasyPolynomial

If f(x) = 2x² - 6x + 3, find f(2).

A) -5

B) -3

C) 1

D) -1

E) 3

Steps

1f(2) = 2×2² - 6×2 + 3

2= 2×4 - 12 + 3 = -1

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

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=3, d=5, find the 12-th term.

A) 46

B) 52

C) 64

D) 70

E) 58

Steps

1aₙ = a₁ + (n-1)d

2a_12 = 3 + 11×5 = 58

Answer: EGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 4ˣ = 64, find x.

A) 3

B) 1

C) 2

D) 4

E) 5

Steps

164 = 4^3

2so x = 3

Answer: AConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 8x + 4 = 0?

A) 4

B) 8

C) 6

D) 10

E) 12

Steps

1Vieta's:sum of roots = -(-8)/1 = 8

2product of roots = 4

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

Q10MediumDivisibility

How many integers from 1 to 38 are divisible by 5?

A) 3

B) 5

C) 7

D) 9

E) 11

Steps

1⌊38 / 5⌋ = 7

Answer: CCount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 9-gon have?

A) 21

B) 24

C) 30

D) 27

E) 33

Steps

1diagonals = n(n-3)/2

2= 9×6/2 = 27

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

Q15MediumCombination

Compute C(9, 3).

A) 66

B) 75

C) 93

D) 102

E) 84

Steps

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

2= 9×8×7/6 = 84

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

Q17MediumPythagorean

Right triangle legs 8 and 15, find the hypotenuse.

A) 17

B) 13

C) 15

D) 19

E) 21

Steps

1c² = 8² + 15² = 64 + 225 = 289

2c = √289 = 17

Answer: APythagorean theorem a²+b²=c²

Q20HardModular Arith

Find the remainder of 3^6 divided by 5.

A) 2

B) 4

C) 3

D) 5

E) 6

Steps

1compute 3^6 mod 5

2simplify step by step (modular arithmetic)

3=4

Answer: BModular arithmetic simplifies large exponents

Q22HardCompleting Sq

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

A) 1

B) 3

C) 5

D) 7

E) 9

Steps

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

2when x = 2 , min at 5

Answer: CCompleting the square for quadratic extrema

Q24HardInclusion-Excl

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

A) 430

B) 485

C) 595

D) 540

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: DInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

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

A) 45

B) 51

C) 63

D) 69

E) 57

Steps

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

2= 49 + 64 - 56 = 57

Answer: ELaw of cosines is key for solving triangles

Q1EasyPolynomial

If f(x) = 2x² - 8x + 3, find f(2).

A) -9

B) -7

C) -3

D) -5

E) -1

Steps

1f(2) = 2×2² - 8×2 + 3

2= 2×4 - 16 + 3 = -5

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

Q3EasyArithmetic Seq

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

A) 46

B) 53

C) 67

D) 74

E) 60

Steps

1aₙ = a₁ + (n-1)d

2a_12 = 5 + 11×5 = 60

Answer: EGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 4ˣ = 64, find x.

A) 3

B) 1

C) 2

D) 4

E) 5

Steps

164 = 4^3

2so x = 3

Answer: AConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 10x + 4 = 0?

A) 6

B) 10

C) 8

D) 12

E) 14

Steps

1Vieta's:sum of roots = -(-10)/1 = 10

2product of roots = 4

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

Q10MediumDivisibility

How many integers from 1 to 40 are divisible by 5?

A) 4

B) 6

C) 8

D) 10

E) 12

Steps

1⌊40 / 5⌋ = 8

Answer: CCount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 11-gon have?

A) 38

B) 41

C) 47

D) 44

E) 50

Steps

1diagonals = n(n-3)/2

2= 11×8/2 = 44

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

Q15MediumCombination

Compute C(11, 3).

A) 131

B) 148

C) 182

D) 199

E) 165

Steps

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

2= 11×10×9/6 = 165

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

Q17MediumPythagorean

Right triangle legs 3 and 4, find the hypotenuse.

A) 5

B) 1

C) 3

D) 7

E) 9

Steps

1c² = 3² + 4² = 9 + 16 = 25

2c = √25 = 5

Answer: APythagorean theorem a²+b²=c²

Q20HardModular Arith

Find the remainder of 3^6 divided by 5.

A) 2

B) 4

C) 3

D) 5

E) 6

Steps

1compute 3^6 mod 5

2simplify step by step (modular arithmetic)

3=4

Answer: BModular arithmetic simplifies large exponents

Q22HardCompleting Sq

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

A) 3

B) 5

C) 7

D) 9

E) 11

Steps

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

2when x = 2 , min at 7

Answer: CCompleting the square for quadratic extrema

Q24HardInclusion-Excl

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

A) 430

B) 485

C) 595

D) 540

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: DInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

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

A) 57

B) 65

C) 81

D) 89

E) 73

Steps

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

2= 81 + 64 - 72 = 73

Answer: ELaw of cosines is key for solving triangles

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2008 AMC12 Sample Problems (12 questions)

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