2023

2023 AMC8 Paper & Solutions Pick

25 questions with step-by-step solutions, topic summaries and method comparisons. Hard combinatorics, similar triangles in geometry.

Sample Problems Download PDF

Scan to get free 2023 PDF + solutions

12 Sample Questions

40 Minutes

Max Score 25

No Calculator

Exam Overview

This exam emphasizes geometry, combinatorics. 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%)

18+

Achievement Roll

Grade 6 and below · 15+ points

15+

Sample Problems

2023 AMC8 Sample Problems (12 questions)

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

Q2EasyPercentage

After a 30% discount, an item sells for 210 yuan. What was the original price?

A) 280

B) 300

C) 290

D) 310

E) 320

Steps

Let the original price be x.

0.7x = 210 → x = 300

Answer: BA 30% discount means the original price × 0.7.

Q4EasyArithmetic

It takes Xiaoming 15 minutes to walk from home to school and 6 minutes to run. If he walks there and runs back, how many minutes does the round trip take in total?

A) 21

B) 18

C) 19

D) 20

E) 22

Steps

Going (walking) = 15 minutes

Returning (running) = 6 minutes

Total = 15 + 6 = 21 minutes

Answer: ABe careful to distinguish the two different modes for the trip there and back.

Q6EasyRatio

A rectangle has a length-to-width ratio of 5:3 and a perimeter of 48 cm. What is its area?

A) 120

B) 150

C) 165

D) 180

E) 135

Steps

Let the length be 5k and the width be 3k.

2(5k+3k) = 48 → 16k = 48 → k = 3

Length = 15, width = 9

Area = 15 × 9 = 135

Answer: ESetting up ratios with k is the standard method for solving ratio problems.

Q8EasyRate / Speed

A car travels at 60 km/h for 2 hours, then at 80 km/h for 1.5 hours. What is the average speed for the whole trip?

A) 65

B) 68 4/7

C) 68

D) 70

E) 72

Steps

Distance = 120 + 120 = 240 km

Time = 3.5 h

Average speed = 240/3.5 = 480/7 ≈ 68 4/7

Answer: BAverage speed = total distance ÷ total time.

Q12MediumGCD/LCM

Find the sum of the greatest common divisor and the least common multiple of 84 and 126.

A) 252

B) 282

C) 306

D) 336

E) 294

Steps

84 = 2² × 3 × 7

126 = 2 × 3² × 7

GCD = 2 × 3 × 7 = 42

LCM = 2² × 3² × 7 = 252

Sum = 42 + 252 = 294

Answer: EGCD × LCM = the product of the two numbers.

Q14MediumCoordinate geometry

What are the coordinates of the midpoint of points A(1,2) and B(5,8)?

A) (2,4)

B) (3,6)

C) (4,5)

D) (4,6)

E) (3,5)

Steps

Midpoint = ((1+5)/2, (2+8)/2) = (3, 5)

Answer: EMidpoint formula: take the average of the two coordinates.

Q16MediumPythagorean Thm.

In isosceles triangle ABC, AB=AC=5 and BC=8. A perpendicular AD is drawn from A to BC. Find AD.

A) 2

B) 4

C) √7

D) 3

E) √11

Steps

D is the midpoint of BC, so BD = 4.

AD² = AB² - BD² = 25 - 16 = 9

AD = 3

Answer: DUse the "three lines coincide" property of an isosceles triangle.

Q19MediumApply the relevant mathematical concept

How many three-digit numbers with no repeated digits can be formed using the four digits 0, 1, 2, 3?

A) 12

B) 16

C) 18

D) 20

E) 24

Steps

The hundreds digit can be 1, 2, or 3 (3 choices).

The tens digit can be any of the remaining 3 digits (3 choices).

The units digit can be any of the remaining 2 digits (2 choices).

Total = 3 × 3 × 2 = 18

Answer: CThe leading digit cannot be 0 — this is the key constraint in this permutation problem.

Q21HardSimilar triangles: proportional sides

Two similar triangles have an area ratio of 4:9. The smaller triangle has a perimeter of 12. What is the perimeter of the larger triangle?

A) 15

B) 16

C) 20

D) 18

E) 24

Steps

Area ratio = 4:9 → side length ratio = 2:3

Perimeter ratio = side length ratio = 2:3

Perimeter of the larger triangle = 12 × 3/2 = 18

Answer: DSimilar triangles: the area ratio equals the square of the side length ratio.

Q22HardDivisor counting

How many numbers from 1 to 50 have exactly 4 positive divisors?

A) 12

B) 13

C) 14

D) 15

E) 16

Steps

4 divisors → p³ or p×q (distinct primes)

p³: 8, 27 (2 numbers)

p×q: 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46 (13 numbers)

Total = 2 + 13 = 15

Answer: DSystematically listing p×q in increasing order of the small prime avoids missing any.

Q24HardCombinatorics

A committee of 3 people is to be chosen from 7 people, with at least one of A and B included. How many ways are there to choose it?

A) 25

B) 20

C) 30

D) 35

E) 40

Steps

Total number of choices: C(7,3) = 35

Neither A nor B is chosen: choose 3 from the other 5 people = C(5,3) = 10

At least one chosen = 35 - 10 = 25

Answer: A"At least" problems are simpler using the complement method.

Q25HardNumber Theory

Find the largest positive integer n among all positive integers n for which n² + 2n + 12 is a perfect square.

A) 3

B) 5

C) 6

D) 8

E) 4

Steps

Let n² + 2n + 12 = k².

k² - (n+1)² = 11

(k-n-1)(k+n+1) = 11 = 1×11

k-n-1=1, k+n+1=11 → n=4, k=6

Check: 16+8+12=36=6² ✓

Answer: ECompleting the square turns the problem into a factoring problem.

Get Full 2023 AMC8 Exam + Solutions

Scan the QR code below to get free 2023 PDF + solutions

2023 AMC8 Paper & Solutions Pick

Scan to get free 2023 PDF + solutions

Exam Overview

DDifficulty

TTopics

AAwards

2023 AMC8 Sample Problems (12 questions)

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Steps

Get Full 2023 AMC8 Exam + Solutions

All Years Navigation

Contact Us