2022

2022 AMC8 Paper & Solutions Pick

First in-person exam since pandemic, 25 solutions. Number theory features Chinese Remainder Theorem variant.

Sample Problems Download PDF

Scan to get free 2022 PDF + solutions

12 Sample Questions

40 Minutes

Max Score 25

No Calculator

Exam Overview

This exam emphasizes number theory. Overall difficulty is moderate.

DDifficulty

EasyQ1-10
MediumQ11-20
HardQ21-25

TTopics

Algebra35%

Geometry30%

Number Theory20%

Combinatorics15%

AAwards

Distinguished Honor Roll

Distinguished Honor Roll (Top 1%)

22+

Honor Roll

Honor Roll (Top 5%)

18+

Achievement Roll

Grade 6 and below · 15+ points

15+

Sample Problems

2022 AMC8 Sample Problems (12 questions)

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

Q1EasyArithmetic

Calculate the value of 1+2+3+...+20.

A) 190

B) 200

C) 210

D) 220

E) 230

Steps

Sum of an arithmetic sequence: n(n+1)/2 = 20×21/2 = 210

Answer: CSum of an arithmetic sequence: S = n(a₁+aₙ)/2

Q4EasyKey concept: ratios

Two people, A and B, have 120 dollars in total. After A gives B 20 dollars, they have equal amounts. How much did A have originally?

A) 60

B) 80

C) 70

D) 90

E) 100

Steps

After giving, each has 60 dollars

A originally had 60 + 20 = 80 dollars

Answer: BA gives B 20 dollars → A decreases by 20, B increases by 20

Q7EasyConsecutive integers

The sum of three consecutive odd numbers is 81. What is the largest of them?

A) 25

B) 27

C) 31

D) 33

E) 29

Steps

Let the middle number be x, then 3x = 81 → x = 27

The largest = 29

Answer: ELet the consecutive odd numbers be x-2, x, x+2

Q9EasyArea

A square garden has an area of 144 m². What is its perimeter?

A) 36

B) 42

C) 56

D) 64

E) 48

Steps

Side length = √144 = 12

Perimeter = 4 × 12 = 48

Answer: EArea of a square = side², perimeter = 4 × side

Q11MediumAlgebra

If 2x + 3y = 12 and x - y = 1, find the value of x.

A) 3

B) 1

C) 2

D) 4

E) 5

Steps

From x = y + 1, substitute

2(y+1) + 3y = 12 → 5y + 2 = 12 → y = 2

x = 3

Answer: ASubstitution is the basic method for solving systems of equations

Q14MediumPythagorean Thm.

A 10-meter ladder leans against a wall with its base 6 meters from the wall. If the base slides out 1 meter, how far does the top slide down?

A) 0.5

B) 1

C) 1.14

D) 2

E) 0.86

Steps

Initial height = √(100-36) = 8

Height after sliding = √(100-49) = √51 ≈ 7.14

Slides down ≈ 0.86

Answer: EKey to ladder problems: the ladder's length stays constant

Q16MediumNumber Theory

A number leaves a remainder of 2 when divided by 3, a remainder of 3 when divided by 5, and a remainder of 2 when divided by 7. What is the smallest such number?

A) 38

B) 23

C) 53

D) 68

E) 83

Steps

Remainder 2 when divided by 3 and remainder 2 when divided by 7 → n = 21k + 2

21k + 2 ≡ 3 (mod 5) → 21k ≡ 1 (mod 5) → k ≡ 1 (mod 5)

k=1: n=23, check 23÷3=7 remainder 2 ✓ 23÷5=4 remainder 3 ✓ 23÷7=3 remainder 2 ✓

Answer: BA classic application of the Chinese Remainder Theorem

Q19MediumApply the relevant mathematical concept

In a class of 40 students, 25 like basketball, 20 like soccer, and 10 like both. How many like neither?

A) 5

B) 3

C) 8

D) 10

E) 15

Steps

Like at least one = 25 + 20 - 10 = 35

Like neither = 40 - 35 = 5

Answer: AInclusion-exclusion principle: |A∪B| = |A| + |B| - |A∩B|

Q21HardNumber Theory

How many numbers from 1 to 200 are divisible by neither 2 nor 5?

A) 60

B) 80

C) 70

D) 90

E) 100

Steps

Divisible by 2: 100 numbers, divisible by 5: 40 numbers

Divisible by 10: 20 numbers

Divisible by 2 or 5 = 100+40-20 = 120

Divisible by neither = 200 - 120 = 80

Answer: BUse the inclusion-exclusion principle for "not divisible" problems

Q22HardGeometry

In trapezoid ABCD, AD∥BC, AD=4, BC=6, and the height is 10. The diagonals intersect at point E. Find the area of triangle ADE.

A) 6

B) 10

C) 12

D) 8

E) 16

Steps

△ADE ∽ △CBE, with ratio of similarity = AD:BC = 4:6 = 2:3

The ratio of the heights from E to AD and BC = 2:3, and their sum = 10 → the height of ADE = 10×2/5 = 4

S(ADE) = 1/2 × AD × height = 1/2 × 4 × 4 = 8

Answer: DThe intersection of a trapezoid's diagonals divides the height in the ratio of the parallel sides

Q24HardDivisibility

Find the smallest positive integer n such that exactly 15 numbers from 1 to n are divisible by 3 but not by 2.

A) 81

B) 85

C) 89

D) 87

E) 91

Steps

Odd numbers divisible by 3: 3, 9, 15, 21, ... (common difference 6)

The 15th = 3+(15-1)×6 = 87

Answer: DThe nth term of an arithmetic sequence = a₁ + (n-1)d

Q25HardProbability

Two people, A and B, each roll a die. What is the probability that A's number is strictly greater than B's?

A) 1/2

B) 7/12

C) 15/36

D) 17/36

E) 5/12

Steps

Total outcomes: 36

Cases where A>B: A=2,B=1 (1); A=3,B=1,2 (2); ...; A=6,B=1-5 (5)

= 1+2+3+4+5 = 15

Probability = 15/36 = 5/12

Answer: EBy symmetry: P(A>B) = P(B>A), and P(equal) = 1/6

Get Full 2022 AMC8 Exam + Solutions

Scan the QR code below to get free 2022 PDF + solutions

2022 AMC8 Paper & Solutions Pick

Scan to get free 2022 PDF + solutions

Exam Overview

DDifficulty

TTopics

AAwards

2022 AMC8 Sample Problems (12 questions)

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Get Full 2022 AMC8 Exam + Solutions

All Years Navigation

Contact Us