2019

2019 AMC8 Paper & Solutions Pick

12 selected sample solutions, algebra focuses on equation applications, geometry covers circle properties. Combinatorics introduces recursion.

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12 Sample Questions

40 Minutes

Max Score 25

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Exam Overview

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

DDifficulty

EasyQ1-10
MediumQ11-20
HardQ21-25

TTopics

Algebra35%

Geometry30%

Number Theory15%

Combinatorics20%

AAwards

Distinguished Honor Roll

Distinguished Honor Roll (Top 1%)

21+

Honor Roll

Honor Roll (Top 5%)

17+

Achievement Roll

Grade 6 and below · 15+ points

15+

Sample Problems

2019 AMC8 Sample Problems (12 questions)

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

Q1EasyArithmetic

Calculate the value of 2019 - 1999 + 20.

A) 30

B) 38

C) 42

D) 40

E) 48

Steps

2019 - 1999 = 20, and 20 + 20 = 40

Answer: DA quick mental computation

Q3EasyKey concept: equations

Xiao Ming buys 3 pencils and 2 notebooks for a total of 26 yuan, and each pencil costs 4 yuan. What is the price of one notebook?

A) 5

B) 6

C) 8

D) 7

E) 9

Steps

3×4 + 2x = 26 → 2x = 14 → x = 7

Answer: DThe key to setting up an equation is finding the equal quantities

Q5EasyGeometry

What is the sum of the interior angles of a regular pentagon?

A) 540

B) 360

C) 480

D) 600

E) 720

Steps

(n-2) × 180° = 3 × 180° = 540°

Answer: AThe sum of the interior angles of an n-sided polygon = (n-2) × 180°

Q8EasyApply the relevant mathematical concept

What is the range of the data set 4, 6, 8, 10, 12?

A) 4

B) 8

C) 6

D) 10

E) 12

Steps

Range = maximum - minimum = 12 - 4 = 8

Answer: BThe range reflects how much the data spreads out

Q11MediumKey concept: equations

In a cage of chickens and rabbits, there are 35 heads and 94 feet in total. How many rabbits are there?

A) 12

B) 10

C) 14

D) 16

E) 18

Steps

Let there be x rabbits and (35-x) chickens

4x + 2(35-x) = 94

2x + 70 = 94 → x = 12

Answer: AThe chickens-and-rabbits problem is a classic application of equations

Q13MediumKey concept: circles

A circle has a radius of 5 cm, and a chord has a length of 8 cm. What is the distance from the center of the circle to the chord?

A) 2

B) 4

C) 3

D) √11

E) √39

Steps

The perpendicular from the center bisects the chord, so half the chord = 4

d² + 16 = 25 → d = 3

Answer: CThe perpendicular from the center to the chord, half the chord, and the radius form a right triangle

Q15MediumDivisibility

Among the numbers from 1 to 100, how many are divisible by 7 but not by 3?

A) 12

B) 10

C) 14

D) 16

E) 18

Steps

Divisible by 7: ⌊100/7⌋ = 14 numbers

Divisible by 21 (divisible by both 7 and 3): ⌊100/21⌋ = 4 numbers

14 − 4 = 10

Answer: BAn application of the inclusion-exclusion principle

Q18MediumKey concept: permutations

Four different books are distributed to 2 people, with each person getting at least one book. How many ways are there to distribute them?

A) 8

B) 10

C) 12

D) 16

E) 14

Steps

Total number of distributions: 2⁴ = 16

Subtract the 2 cases where all books go to one person

16 - 2 = 14

Answer: ECount all arrangements first, then subtract the ones that don't meet the condition

Q21HardFind the recursive pattern

f(1)=1, f(2)=3, f(n)=f(n-1)+f(n-2). Find f(7).

A) 17

B) 19

C) 29

D) 21

E) 23

Steps

f(3)=4, f(4)=7, f(5)=11, f(6)=18, f(7)=29

Answer: CComputing term by term using the recurrence is the basic method for sequence problems

Q22HardArea

An isosceles right triangle has a hypotenuse of 10. What is its area?

A) 20

B) 30

C) 40

D) 50

E) 25

Steps

Let each leg be a, so a² + a² = 100

2a² = 100 → a² = 50

Area = a²/2 = 25

Answer: EThe area of an isosceles right triangle = hypotenuse²/4

Q24HardCombinatorics

Choosing 4 different numbers from {1, 2, …, 8} so that their sum is even, how many ways are there?

A) 30

B) 33

C) 36

D) 38

E) 46

Steps

Among 1-8 there are 4 odd numbers and 4 even numbers

For the sum to be even: 4 even / 4 odd / 2 even and 2 odd

4 even: C(4,4)=1; 4 odd: C(4,4)=1

2 even and 2 odd: C(4,2)×C(4,2)=6×6=36

Total: 1+1+36 = 38

Answer: DThe sum of four numbers is even when they are all even, all odd, or two odd and two even

Q25HardNumber Theory

Find the positive integer n satisfying 1/n + 1/(n+1) = 5/6.

A) 1

B) 3

C) 4

D) 2

E) 5

Steps

Combining the fractions: (2n+1)/(n(n+1)) = 5/6

6(2n+1) = 5n(n+1)

12n+6 = 5n²+5n

5n² - 7n - 6 = 0

(5n+3)(n-2)=0 → n=2

Answer: DAfter combining the fractions, the equation becomes a quadratic equation

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2019 AMC8 Paper & Solutions Pick

Scan to get free 2019 PDF + solutions

Exam Overview

DDifficulty

TTopics

AAwards

2019 AMC8 Sample Problems (12 questions)

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