Paper Set

2005

2005 AMC10 Paper & Solutions Pick Paper A

25 questions with solutions, moderate difficulty, balanced topic distribution.

Sample Problems Download PDF

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25 Questions

40 Minutes

Max Score 25

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Exam Overview

This exam emphasizes multiple math topics. 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%)

108+

Honor Roll

Honor Roll (Top 5%)

98+

Achievement Roll

Grade 6 and below · 15+ points

90+

Sample Problems

2005 AMC10 Sample Problems (12 questions)

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

Q1EasyPolynomial

If f(x) = 2x² - 3x + 3, find f(3).

A) 12

B) 8

C) 10

D) 14

E) 16

Steps

1f(3) = 2×3² - 3×3 + 3

2= 2×9 - 9 + 3 = 12

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

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=4, d=2, find the 15-th term.

A) 24

B) 32

C) 28

D) 36

E) 40

Steps

1aₙ = a₁ + (n-1)d

2a_15 = 4 + 14×2 = 32

Answer: BGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 4ˣ = 256, find x.

A) 2

B) 3

C) 4

D) 5

E) 6

Steps

1256 = 4^4

2so x = 4

Answer: CConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 5x + 5 = 0?

A) 1

B) 3

C) 7

D) 5

E) 9

Steps

1Vieta's:sum of roots = -(-5)/1 = 5

2product of roots = 5

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

Q10MediumDivisibility

How many integers from 1 to 35 are divisible by 5?

A) 3

B) 5

C) 9

D) 11

E) 7

Steps

1⌊35 / 5⌋ = 7

Answer: ECount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 12-gon have?

A) 54

B) 48

C) 51

D) 57

E) 60

Steps

1diagonals = n(n-3)/2

2= 12×9/2 = 54

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

Q15MediumCombination

Compute C(6, 3).

A) 14

B) 20

C) 17

D) 23

E) 26

Steps

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

2= 6×5×4/6 = 20

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

Q17MediumPythagorean

Right triangle legs 3 and 4, find the hypotenuse.

A) 1

B) 3

C) 5

D) 7

E) 9

Steps

1c² = 3² + 4² = 9 + 16 = 25

2c = √25 = 5

Answer: CPythagorean theorem a²+b²=c²

Q20HardModular Arith

Find the remainder of 4^6 divided by 6.

A) 2

B) 3

C) 5

D) 4

E) 6

Steps

1compute 4^6 mod 6

2simplify step by step (modular arithmetic)

3=4

Answer: DModular arithmetic simplifies large exponents

Q22HardCompleting Sq

Find the minimum of f(x) = x² - 6x + 14.

A) 1

B) 3

C) 7

D) 9

E) 5

Steps

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

2when x = 3 , min at 5

Answer: ECompleting the square for quadratic extrema

Q24HardInclusion-Excl

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

A) 540

B) 430

C) 485

D) 595

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: AInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

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

A) 45

B) 57

C) 51

D) 63

E) 69

Steps

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

2= 49 + 64 - 56 = 57

Answer: BLaw of cosines is key for solving triangles

Q1EasyPolynomial

If f(x) = 2x² - 5x + 3, find f(3).

A) 6

B) 2

C) 4

D) 8

E) 10

Steps

1f(3) = 2×3² - 5×3 + 3

2= 2×9 - 15 + 3 = 6

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

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=6, d=2, find the 15-th term.

A) 26

B) 34

C) 30

D) 38

E) 42

Steps

1aₙ = a₁ + (n-1)d

2a_15 = 6 + 14×2 = 34

Answer: BGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 4ˣ = 256, find x.

A) 2

B) 3

C) 4

D) 5

E) 6

Steps

1256 = 4^4

2so x = 4

Answer: CConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 7x + 5 = 0?

A) 3

B) 5

C) 9

D) 7

E) 11

Steps

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

2product of roots = 5

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

Q10MediumDivisibility

How many integers from 1 to 37 are divisible by 5?

A) 3

B) 5

C) 9

D) 11

E) 7

Steps

1⌊37 / 5⌋ = 7

Answer: ECount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 14-gon have?

A) 77

B) 71

C) 74

D) 80

E) 83

Steps

1diagonals = n(n-3)/2

2= 14×11/2 = 77

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

Q15MediumCombination

Compute C(8, 3).

A) 44

B) 56

C) 50

D) 62

E) 68

Steps

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

2= 8×7×6/6 = 56

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

Q17MediumPythagorean

Right triangle legs 6 and 8, find the hypotenuse.

A) 6

B) 8

C) 10

D) 12

E) 14

Steps

1c² = 6² + 8² = 36 + 64 = 100

2c = √100 = 10

Answer: CPythagorean theorem a²+b²=c²

Q20HardModular Arith

Find the remainder of 4^6 divided by 6.

A) 2

B) 3

C) 5

D) 4

E) 6

Steps

1compute 4^6 mod 6

2simplify step by step (modular arithmetic)

3=4

Answer: DModular arithmetic simplifies large exponents

Q22HardCompleting Sq

Find the minimum of f(x) = x² - 6x + 16.

A) 3

B) 5

C) 9

D) 11

E) 7

Steps

1complete the square:f(x) = (x - 3)² + 7

2when x = 3 , min at 7

Answer: ECompleting the square for quadratic extrema

Q24HardInclusion-Excl

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

A) 540

B) 430

C) 485

D) 595

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: AInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

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

A) 57

B) 73

C) 65

D) 81

E) 89

Steps

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

2= 81 + 64 - 72 = 73

Answer: BLaw of cosines is key for solving triangles

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2005 AMC10 Paper & Solutions Pick Paper A

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2005 AMC10 Sample Problems (12 questions)

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