Paper Set

2015

2015 AMC10 Paper & Solutions Pick Paper A

25 questions with solutions, geometry focuses on triangle and quadrilateral areas, algebra on systems of equations.

Sample Problems Download PDF

Scan to get free 2015 PDF + solutions

25 Questions

40 Minutes

Max Score 25

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Exam Overview

This exam emphasizes algebra, geometry, equations. 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%)

105+

Honor Roll

Honor Roll (Top 5%)

100+

Achievement Roll

Grade 6 and below · 15+ points

88+

Sample Problems

2015 AMC10 Sample Problems (12 questions)

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

Q1EasyPolynomial

If f(x) = 2x² - 3x + 1, find f(5).

A) 36

B) 26

C) 31

D) 41

E) 46

Steps

1f(5) = 2×5² - 3×5 + 1

2= 2×25 - 15 + 1 = 36

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

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=6, d=2, find the 13-th term.

A) 22

B) 30

C) 26

D) 34

E) 38

Steps

1aₙ = a₁ + (n-1)d

2a_13 = 6 + 12×2 = 30

Answer: BGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 2ˣ = 64, find x.

A) 4

B) 5

C) 6

D) 7

E) 8

Steps

164 = 2^6

2so x = 6

Answer: CConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 5x + 7 = 0?

A) 1

B) 3

C) 7

D) 5

E) 9

Steps

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

2product of roots = 7

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

Q10MediumDivisibility

How many integers from 1 to 45 are divisible by 3?

A) 11

B) 13

C) 17

D) 19

E) 15

Steps

1⌊45 / 3⌋ = 15

Answer: ECount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 10-gon have?

A) 35

B) 29

C) 32

D) 38

E) 41

Steps

1diagonals = n(n-3)/2

2= 10×7/2 = 35

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 6^4 divided by 8.

A) 1

B) 2

C) 3

D) 0

E) 4

Steps

1compute 6^4 mod 8

2simplify step by step (modular arithmetic)

3=0

Answer: DModular arithmetic simplifies large exponents

Q22HardCompleting Sq

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

A) 1

B) 5

C) 7

D) 9

E) 3

Steps

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

2when x = 5 , min at 3

Answer: ECompleting the square for quadratic extrema

Q24HardInclusion-Excl

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

A) 36

B) 26

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: AInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

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

A) 23

B) 31

C) 27

D) 35

E) 39

Steps

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

2= 25 + 36 - 30 = 31

Answer: BLaw of cosines is key for solving triangles

Q1EasyPolynomial

If f(x) = 2x² - 5x + 1, find f(5).

A) 26

B) 18

C) 22

D) 30

E) 34

Steps

1f(5) = 2×5² - 5×5 + 1

2= 2×25 - 25 + 1 = 26

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

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=8, d=2, find the 13-th term.

A) 24

B) 32

C) 28

D) 36

E) 40

Steps

1aₙ = a₁ + (n-1)d

2a_13 = 8 + 12×2 = 32

Answer: BGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 2ˣ = 64, find x.

A) 4

B) 5

C) 6

D) 7

E) 8

Steps

164 = 2^6

2so x = 6

Answer: CConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 7x + 7 = 0?

A) 3

B) 5

C) 9

D) 7

E) 11

Steps

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

2product of roots = 7

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

Q10MediumDivisibility

How many integers from 1 to 47 are divisible by 3?

A) 11

B) 13

C) 17

D) 19

E) 15

Steps

1⌊47 / 3⌋ = 15

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(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 6^4 divided by 8.

A) 1

B) 2

C) 3

D) 0

E) 4

Steps

1compute 6^4 mod 8

2simplify step by step (modular arithmetic)

3=0

Answer: DModular arithmetic simplifies large exponents

Q22HardCompleting Sq

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

A) 1

B) 3

C) 7

D) 9

E) 5

Steps

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

2when x = 5 , min at 5

Answer: ECompleting the square for quadratic extrema

Q24HardInclusion-Excl

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

A) 36

B) 26

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: AInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

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

A) 33

B) 43

C) 38

D) 48

E) 53

Steps

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

2= 49 + 36 - 42 = 43

Answer: BLaw of cosines is key for solving triangles

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2015 AMC10 Sample Problems (12 questions)

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