Paper Set

2006

2006 AMC12 Paper & Solutions Pick Paper A

25 questions with solutions, geometry focuses on circles and triangles, practical word problems.

Sample Problems Download PDF

Scan to get free 2006 PDF + solutions

25 Questions

75 Minutes

Max Score 150

No Calculator

Exam Overview

This exam emphasizes geometry. Overall difficulty is moderate.

DDifficulty

EasyQ1-10
MediumQ11-20
HardQ21-25

TTopics

Algebra35%

Geometry30%

Number Theory15%

Combinatorics20%

AAwards

AIME Qualification

AIME Qualification (Top 2.5%)

91+

Honor Roll

Honor Roll (Top 5%)

81+

Achievement Roll

Grade 10 and below

60+

Sample Problems

2006 AMC12 Sample Problems (12 questions)

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

Q1EasyPolynomial

If f(x) = 2x² - 4x + 1, find f(4).

A) 11

B) 17

C) 14

D) 20

E) 23

Steps

1f(4) = 2×4² - 4×4 + 1

2= 2×16 - 16 + 1 = 17

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

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=5, d=3, find the 10-th term.

A) 24

B) 28

C) 32

D) 36

E) 40

Steps

1aₙ = a₁ + (n-1)d

2a_10 = 5 + 9×3 = 32

Answer: CGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 2ˣ = 32, find x.

A) 3

B) 4

C) 6

D) 5

E) 7

Steps

132 = 2^5

2so x = 5

Answer: DConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 6x + 6 = 0?

A) 2

B) 4

C) 8

D) 10

E) 6

Steps

1Vieta's:sum of roots = -(-6)/1 = 6

2product of roots = 6

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

Q10MediumDivisibility

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

A) 12

B) 8

C) 10

D) 14

E) 16

Steps

1⌊36 / 3⌋ = 12

Answer: ACount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 7-gon have?

A) 8

B) 14

C) 11

D) 17

E) 20

Steps

1diagonals = n(n-3)/2

2= 7×4/2 = 14

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

Q15MediumCombination

Compute C(7, 3).

A) 27

B) 31

C) 35

D) 39

E) 43

Steps

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

2= 7×6×5/6 = 35

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

Q17MediumPythagorean

Right triangle legs 5 and 12, find the hypotenuse.

A) 9

B) 11

C) 15

D) 13

E) 17

Steps

1c² = 5² + 12² = 25 + 144 = 169

2c = √169 = 13

Answer: DPythagorean theorem a²+b²=c²

Q20HardModular Arith

Find the remainder of 5^4 divided by 7.

A) 0

B) 1

C) 3

D) 4

E) 2

Steps

1compute 5^4 mod 7

2simplify step by step (modular arithmetic)

3=2

Answer: EModular arithmetic simplifies large exponents

Q22HardCompleting Sq

Find the minimum of f(x) = x² - 8x + 19.

A) 3

B) 1

C) 5

D) 7

E) 9

Steps

1complete the square:f(x) = (x - 4)² + 3

2when x = 4 , min at 3

Answer: ACompleting the square for quadratic extrema

Q24HardInclusion-Excl

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

A) 26

B) 36

C) 31

D) 41

E) 46

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: BInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

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

A) 23

B) 27

C) 31

D) 35

E) 39

Steps

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

2= 25 + 36 - 30 = 31

Answer: CLaw of cosines is key for solving triangles

Q1EasyPolynomial

If f(x) = 2x² - 6x + 1, find f(4).

A) 5

B) 9

C) 7

D) 11

E) 13

Steps

1f(4) = 2×4² - 6×4 + 1

2= 2×16 - 24 + 1 = 9

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

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=7, d=3, find the 10-th term.

A) 26

B) 30

C) 34

D) 38

E) 42

Steps

1aₙ = a₁ + (n-1)d

2a_10 = 7 + 9×3 = 34

Answer: CGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 2ˣ = 32, find x.

A) 3

B) 4

C) 6

D) 5

E) 7

Steps

132 = 2^5

2so x = 5

Answer: DConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 8x + 6 = 0?

A) 4

B) 6

C) 10

D) 12

E) 8

Steps

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

2product of roots = 6

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

Q10MediumDivisibility

How many integers from 1 to 38 are divisible by 3?

A) 12

B) 8

C) 10

D) 14

E) 16

Steps

1⌊38 / 3⌋ = 12

Answer: ACount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 9-gon have?

A) 21

B) 27

C) 24

D) 30

E) 33

Steps

1diagonals = n(n-3)/2

2= 9×6/2 = 27

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

Q15MediumCombination

Compute C(9, 3).

A) 66

B) 75

C) 84

D) 93

E) 102

Steps

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

2= 9×8×7/6 = 84

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

Q17MediumPythagorean

Right triangle legs 8 and 15, find the hypotenuse.

A) 13

B) 15

C) 19

D) 17

E) 21

Steps

1c² = 8² + 15² = 64 + 225 = 289

2c = √289 = 17

Answer: DPythagorean theorem a²+b²=c²

Q20HardModular Arith

Find the remainder of 5^4 divided by 7.

A) 0

B) 1

C) 3

D) 4

E) 2

Steps

1compute 5^4 mod 7

2simplify step by step (modular arithmetic)

3=2

Answer: EModular arithmetic simplifies large exponents

Q22HardCompleting Sq

Find the minimum of f(x) = x² - 8x + 21.

A) 5

B) 1

C) 3

D) 7

E) 9

Steps

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

2when x = 4 , min at 5

Answer: ACompleting the square for quadratic extrema

Q24HardInclusion-Excl

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

A) 26

B) 36

C) 31

D) 41

E) 46

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: BInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

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

A) 33

B) 38

C) 43

D) 48

E) 53

Steps

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

2= 49 + 36 - 42 = 43

Answer: CLaw of cosines is key for solving triangles

Get Full 2006 AMC12 Exam + Solutions

Scan the QR code below to get free 2006 PDF + solutions

All Years

All Years Navigation

All AMC12 exam years from 2000-2025, click to view

2006 AMC12 Paper & Solutions Pick Paper A

Scan to get free 2006 PDF + solutions

Exam Overview

DDifficulty

TTopics

AAwards

2006 AMC12 Sample Problems (12 questions)

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Get Full 2006 AMC12 Exam + Solutions

All Years Navigation

Contact Us