Paper Set

2003

2003 AMC12 Paper & Solutions Pick Paper A

25 questions with solutions, algebra focuses on equations, geometry covers basic shapes.

Sample Problems Download PDF

Scan to get free 2003 PDF + solutions

25 Questions

75 Minutes

Max Score 150

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Exam Overview

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

DDifficulty

EasyQ1-10
MediumQ11-20
HardQ21-25

TTopics

Algebra40%

Geometry30%

Number Theory15%

Combinatorics15%

AAwards

AIME Qualification

AIME Qualification (Top 2.5%)

88+

Honor Roll

Honor Roll (Top 5%)

78+

Achievement Roll

Grade 10 and below

63+

Sample Problems

2003 AMC12 Sample Problems (12 questions)

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

Q1EasyPolynomial

If f(x) = 2x² - 6x + 1, find f(5).

A) 15

B) 18

C) 24

D) 21

E) 27

Steps

1f(5) = 2×5² - 6×5 + 1

2= 2×25 - 30 + 1 = 21

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

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=6, d=5, find the 13-th term.

A) 52

B) 59

C) 73

D) 80

E) 66

Steps

1aₙ = a₁ + (n-1)d

2a_13 = 6 + 12×5 = 66

Answer: EGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 2ˣ = 64, find x.

A) 6

B) 4

C) 5

D) 7

E) 8

Steps

164 = 2^6

2so x = 6

Answer: AConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 8x + 7 = 0?

A) 4

B) 8

C) 6

D) 10

E) 12

Steps

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

2product of roots = 7

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

Q10MediumDivisibility

How many integers from 1 to 33 are divisible by 3?

A) 7

B) 9

C) 11

D) 13

E) 15

Steps

1⌊33 / 3⌋ = 11

Answer: CCount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 10-gon have?

A) 29

B) 32

C) 38

D) 35

E) 41

Steps

1diagonals = n(n-3)/2

2= 10×7/2 = 35

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 6^4 divided by 8.

A) 1

B) 0

C) 2

D) 3

E) 4

Steps

1compute 6^4 mod 8

2simplify step by step (modular arithmetic)

3=0

Answer: BModular arithmetic simplifies large exponents

Q22HardCompleting Sq

Find the minimum of f(x) = x² - 10x + 28.

A) 1

B) 5

C) 3

D) 7

E) 9

Steps

1complete the square:f(x) = (x - 5)² + 3

2when x = 5 , min at 3

Answer: CCompleting 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) 36

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: DInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

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

A) 23

B) 27

C) 35

D) 39

E) 31

Steps

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

2= 25 + 36 - 30 = 31

Answer: ELaw of cosines is key for solving triangles

Q1EasyPolynomial

If f(x) = 2x² - 8x + 1, find f(5).

A) 7

B) 9

C) 13

D) 11

E) 15

Steps

1f(5) = 2×5² - 8×5 + 1

2= 2×25 - 40 + 1 = 11

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

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=8, d=5, find the 13-th term.

A) 54

B) 61

C) 75

D) 82

E) 68

Steps

1aₙ = a₁ + (n-1)d

2a_13 = 8 + 12×5 = 68

Answer: EGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 2ˣ = 64, find x.

A) 6

B) 4

C) 5

D) 7

E) 8

Steps

164 = 2^6

2so x = 6

Answer: AConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 10x + 7 = 0?

A) 6

B) 10

C) 8

D) 12

E) 14

Steps

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

2product of roots = 7

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

Q10MediumDivisibility

How many integers from 1 to 35 are divisible by 3?

A) 7

B) 9

C) 11

D) 13

E) 15

Steps

1⌊35 / 3⌋ = 11

Answer: CCount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 12-gon have?

A) 48

B) 51

C) 57

D) 54

E) 60

Steps

1diagonals = n(n-3)/2

2= 12×9/2 = 54

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 6^4 divided by 8.

A) 1

B) 0

C) 2

D) 3

E) 4

Steps

1compute 6^4 mod 8

2simplify step by step (modular arithmetic)

3=0

Answer: BModular arithmetic simplifies large exponents

Q22HardCompleting Sq

Find the minimum of f(x) = x² - 10x + 30.

A) 1

B) 3

C) 5

D) 7

E) 9

Steps

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

2when x = 5 , min at 5

Answer: CCompleting 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) 36

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: DInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

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

A) 33

B) 38

C) 48

D) 53

E) 43

Steps

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

2= 49 + 36 - 42 = 43

Answer: ELaw of cosines is key for solving triangles

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2003 AMC12 Sample Problems (12 questions)

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