Paper Set

2007

2007 AMC12 Paper & Solutions Pick Paper A

25 questions with solutions, significant number theory and geometry weight.

Sample Problems Download PDF

Scan to get free 2007 PDF + solutions

25 Questions

75 Minutes

Max Score 150

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Exam Overview

This exam emphasizes geometry, number theory. Overall difficulty is moderate.

DDifficulty

EasyQ1-10
MediumQ11-20
HardQ21-25

TTopics

Algebra30%

Geometry35%

Number Theory20%

Combinatorics15%

AAwards

AIME Qualification

AIME Qualification (Top 2.5%)

92+

Honor Roll

Honor Roll (Top 5%)

82+

Achievement Roll

Grade 10 and below

61+

Sample Problems

2007 AMC12 Sample Problems (12 questions)

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

Q1EasyPolynomial

If f(x) = 2x² - 5x + 2, find f(5).

A) 19

B) 23

C) 27

D) 31

E) 35

Steps

1f(5) = 2×5² - 5×5 + 2

2= 2×25 - 25 + 2 = 27

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

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=6, d=4, find the 11-th term.

A) 36

B) 41

C) 51

D) 46

E) 56

Steps

1aₙ = a₁ + (n-1)d

2a_11 = 6 + 10×4 = 46

Answer: DGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 3ˣ = 729, find x.

A) 4

B) 5

C) 7

D) 8

E) 6

Steps

1729 = 3^6

2so x = 6

Answer: EConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 7x + 7 = 0?

A) 7

B) 3

C) 5

D) 9

E) 11

Steps

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

2product of roots = 7

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

Q10MediumDivisibility

How many integers from 1 to 37 are divisible by 4?

A) 5

B) 9

C) 7

D) 11

E) 13

Steps

1⌊37 / 4⌋ = 9

Answer: BCount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 8-gon have?

A) 14

B) 17

C) 20

D) 23

E) 26

Steps

1diagonals = n(n-3)/2

2= 8×5/2 = 20

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

Q15MediumCombination

Compute C(8, 3).

A) 44

B) 50

C) 62

D) 56

E) 68

Steps

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

2= 8×7×6/6 = 56

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

Q17MediumPythagorean

Right triangle legs 6 and 8, find the hypotenuse.

A) 6

B) 8

C) 12

D) 14

E) 10

Steps

1c² = 6² + 8² = 36 + 64 = 100

2c = √100 = 10

Answer: EPythagorean theorem a²+b²=c²

Q20HardModular Arith

Find the remainder of 6^5 divided by 8.

A) 0

B) 1

C) 2

D) 3

E) 4

Steps

1compute 6^5 mod 8

2simplify step by step (modular arithmetic)

3=0

Answer: AModular arithmetic simplifies large exponents

Q22HardCompleting Sq

Find the minimum of f(x) = x² - 10x + 29.

A) 0

B) 4

C) 2

D) 6

E) 8

Steps

1complete the square:f(x) = (x - 5)² + 4

2when x = 5 , min at 4

Answer: BCompleting 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) 150

D) 166

E) 182

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: CInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

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

A) 33

B) 38

C) 48

D) 43

E) 53

Steps

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

2= 36 + 49 - 42 = 43

Answer: DLaw of cosines is key for solving triangles

Q1EasyPolynomial

If f(x) = 2x² - 7x + 2, find f(5).

A) 11

B) 14

C) 17

D) 20

E) 23

Steps

1f(5) = 2×5² - 7×5 + 2

2= 2×25 - 35 + 2 = 17

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

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=8, d=4, find the 11-th term.

A) 38

B) 43

C) 53

D) 48

E) 58

Steps

1aₙ = a₁ + (n-1)d

2a_11 = 8 + 10×4 = 48

Answer: DGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 3ˣ = 729, find x.

A) 4

B) 5

C) 7

D) 8

E) 6

Steps

1729 = 3^6

2so x = 6

Answer: EConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 9x + 7 = 0?

A) 9

B) 5

C) 7

D) 11

E) 13

Steps

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

2product of roots = 7

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

Q10MediumDivisibility

How many integers from 1 to 39 are divisible by 4?

A) 5

B) 9

C) 7

D) 11

E) 13

Steps

1⌊39 / 4⌋ = 9

Answer: BCount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 10-gon have?

A) 29

B) 32

C) 35

D) 38

E) 41

Steps

1diagonals = n(n-3)/2

2= 10×7/2 = 35

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

Q15MediumCombination

Compute C(10, 3).

A) 94

B) 107

C) 133

D) 120

E) 146

Steps

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

2= 10×9×8/6 = 120

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

Q17MediumPythagorean

Right triangle legs 7 and 24, find the hypotenuse.

A) 21

B) 23

C) 27

D) 29

E) 25

Steps

1c² = 7² + 24² = 49 + 576 = 625

2c = √625 = 25

Answer: EPythagorean theorem a²+b²=c²

Q20HardModular Arith

Find the remainder of 6^5 divided by 8.

A) 0

B) 1

C) 2

D) 3

E) 4

Steps

1compute 6^5 mod 8

2simplify step by step (modular arithmetic)

3=0

Answer: AModular arithmetic simplifies large exponents

Q22HardCompleting Sq

Find the minimum of f(x) = x² - 10x + 31.

A) 2

B) 6

C) 4

D) 8

E) 10

Steps

1complete the square:f(x) = (x - 5)² + 6

2when x = 5 , min at 6

Answer: BCompleting 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) 150

D) 166

E) 182

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: CInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

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

A) 45

B) 51

C) 63

D) 57

E) 69

Steps

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

2= 64 + 49 - 56 = 57

Answer: DLaw of cosines is key for solving triangles

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2007 AMC12 Sample Problems (12 questions)

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