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2021

2021 AMC8 Paper & Solutions Pick

Online exam year, 12 selected sample solutions. Many practical application problems (shopping discounts, trip planning).

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12 Sample Questions
40 Minutes
Max Score 25
No Calculator
Exam Overview

Exam Overview

This exam emphasizes multiple math topics. Overall difficulty is moderate.

DDifficulty

  • EasyQ1-10
  • MediumQ11-20
  • HardQ21-25

TTopics

  • Algebra40%
  • Geometry25%
  • Number Theory15%
  • Combinatorics20%

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

2021 AMC8 Sample Problems (12 questions)

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

Q2EasyOperations
Calculate the value of 25 × 16 × 4.
A) 1200
B) 1400
C) 1800
D) 2000
E) 1600

Steps

25 × 4 = 100, 100 × 16 = 1600
Or 25 × 16 = 400, 400 × 4 = 1600
Answer: EMaking round numbers is the key technique for fast calculation
Q4EasyConvert percentages to fractions
An item originally costs 80 yuan. Its price is first raised by 25% and then discounted to 80% of the new price. How much does the final price differ from the original price?
A) Same / Equal
B) height5
C) height2
D) 2
E) 5

Steps

Price increase: 80 × 1.25 = 100; 20% off: 100 × 0.8 = 80
Final = original price, a difference of 0 yuan
Answer: ARaising the price first and then lowering it does not necessarily return to the original price
Q6EasyKey concept: equations
Three times a number minus 7 equals two times the number plus 5. What is the number?
A) 6
B) 12
C) 8
D) 10
E) 14

Steps

3x - 7 = 2x + 5
x = 12
Answer: BMove terms across and combine like terms
Q10EasyArea
A triangle has a base of 8 cm and a height of 6 cm. What is its area?
A) 12
B) 28
C) 36
D) 48
E) 24

Steps

Area = 1/2 × 8 × 6 = 24
Answer: EArea of a triangle = base × height ÷ 2
Q11MediumEquation App.
Two places, A and B, are 240 km apart. A passenger bus leaves from A and a freight truck leaves from B, traveling toward each other. The bus travels at 60 km/h and the truck at 40 km/h. How many hours until they meet?
A) 2
B) 3
C) 3.6
D) 4
E) 2.4

Steps

Sum of speeds = 60 + 40 = 100 km/h
Time = 240/100 = 2.4 hours
Answer: EFor objects moving toward each other, use the sum of their speeds
Q13MediumPerfect square
From 1 to 100, how many numbers are both perfect squares and even?
A) 5
B) 3
C) 4
D) 6
E) 7

Steps

An even perfect square = (2k)²
Within 100: 4, 16, 36, 64, 100
5 in total
Answer: AThe square of an even number is always even
Q15MediumDivisibility
From 1 to 30, how many numbers are divisible by 2 or 3?
A) 20
B) 15
C) 18
D) 22
E) 25

Steps

Divisible by 2: 15 numbers; divisible by 3: 10 numbers
Divisible by 6: 5 numbers
Divisible by 2 or 3 = 15 + 10 - 5 = 20
Answer: AInclusion-exclusion principle
Q18MediumProbability
A coin is tossed 3 times. What is the probability of getting exactly 2 heads?
A) 1/8
B) 3/8
C) 1/4
D) 1/2
E) 5/8

Steps

Total outcomes 2³ = 8
Exactly 2 heads: C(3,2) = 3 (HHT, HTH, THH)
Probability = 3/8
Answer: BBinomial probability C(n,k) × p^k × (1-p)^(n-k)
Q21HardGeometry
A regular hexagon has a side length of 4. What is its area? (Use √3 ≈ 1.73)
A) 24√3
B) 36
C) 41.5
D) 48
E) 72

Steps

A regular hexagon = 6 equilateral triangles
Each area = (√3/4) × 4² = 4√3
Total area = 24√3 ≈ 41.56
Answer: AArea of a regular hexagon = (3√3/2) × side length²
Q23HardNumber Theory
Find the units digit of 1! + 2! + 3! + ... + 10!.
A) 1
B) 5
C) 3
D) 7
E) 9

Steps

5! and beyond have a units digit of 0
Only need to compute 1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33
Units digit = 3
Answer: CWhen n ≥ 5, the units digit of n! is 0
Q24HardCombinatorics
Choose 3 distinct numbers from 1 to 20 so that their sum is even. How many ways are there?
A) 500
B) 520
C) 540
D) 570
E) 560

Steps

From 1 to 20 there are 10 odd numbers and 10 even numbers
Sum of three numbers is even: all even, or two odd and one even
All even: C(10,3) = 120
Two odd and one even: C(10,2) × C(10,1) = 45 × 10 = 450
Total: 120 + 450 = 570
Answer: DThe sum of three numbers is even when all are even, or two are odd and one is even
Q25HardProbability
A bag contains 2 red and 3 blue balls. Two balls are drawn. What is the probability of getting exactly 1 red and 1 blue?
A) 2/5
B) 1/2
C) 3/5
D) 2/3
E) 3/4

Steps

C(2,1) × C(3,1) / C(5,2) = 6/10 = 3/5
Answer: CCombination probability = favorable ways ÷ total ways

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