AMC 8 Master Prep — 20 Essential Problems

01

Arithmetic & Order of Operations

The foundation of all AMC 8 problems. Mastering PEMDAS and integer arithmetic prevents careless errors.

▸ Key Facts to Memorize

◆PEMDAS: Parentheses → Exponents → Multiply/Divide (L to R) → Add/Subtract (L to R)

◆Negative × Negative = Positive; Negative × Positive = Negative

◆Division by zero is undefined

3 + 4 × 2² - (6 ÷ 3)
= 3 + 4 × 4 - 2
= 3 + 16 - 2 = 17

▸ Quick Example

What is 5 + 3 × (2² - 1)?
= 5 + 3 × (4 - 1) = 5 + 3 × 3 = 5 + 9 = 14

Answer: 14

02

Fractions, Decimals & Percents

Converting fluidly between forms and computing percent change are high-frequency AMC 8 skills.

▸ Key Facts to Memorize

◆a/b ÷ c/d = a/b × d/c (multiply by reciprocal)

◆Percent change = (New - Old) / Old × 100%

◆1/8 = 0.125, 1/6 ≈ 0.1667, 1/3 ≈ 0.3333

Percent Change = (New − Old) / Old × 100

▸ Quick Example

A price rises from $40 to $50. What is the percent increase?
(50 − 40) / 40 × 100 = 10/40 × 100 = 25%

Answer: 25%

03

Number Theory (GCD, LCM, Primes)

Prime factorization unlocks GCD, LCM, divisor counting, and many AMC 8 shortcuts.

▸ Key Facts to Memorize

◆GCD × LCM = product of the two numbers

◆Number of divisors of p^a × q^b = (a+1)(b+1)

◆Primes to 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

◆A number is divisible by 9 if its digit sum is divisible by 9

GCD(12,18): 12=2²×3, 18=2×3² → GCD=2×3=6
LCM(12,18) = 12×18/6 = 36

▸ Quick Example

How many positive divisors does 36 have?
36 = 2² × 3² → (2+1)(2+1) = 9 divisors

Answer: 9

04

Ratios, Rates & Proportions

Ratios appear in mixture problems, speed/distance, and scaling. Always use a multiplier.

▸ Key Facts to Memorize

◆If a:b = 3:5, let a = 3k, b = 5k, total = 8k

◆Distance = Rate × Time

◆Average speed ≠ arithmetic mean of speeds (use total distance / total time)

Speed = Distance / Time
Average Speed = Total Distance / Total Time

▸ Quick Example

Boys to girls = 3:4, total 28 students. How many boys?
3k + 4k = 7k = 28, so k = 4. Boys = 3×4 = 12

Answer: 12

05

Basic Algebra & Linear Equations

Setting up equations from word problems is the most important AMC 8 algebraic skill.

▸ Key Facts to Memorize

◆Consecutive integers: n, n+1, n+2, ... ; consecutive evens: n, n+2, n+4, ...

◆(a+b)² = a² + 2ab + b²

◆(a-b)(a+b) = a² - b²

(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
(a + b)(a − b) = a² − b²

▸ Quick Example

Three consecutive integers sum to 33. Find the largest.
n + (n+1) + (n+2) = 33 → 3n+3 = 33 → n = 10. Largest = 12

Answer: 12

06

Geometry: Area & Perimeter

Know all standard area formulas cold. Many AMC 8 geometry problems use composite figures.

▸ Key Facts to Memorize

◆Rectangle: A = l×w, P = 2(l+w)

◆Triangle: A = ½ × b × h

◆Circle: A = πr², C = 2πr

◆Trapezoid: A = ½(b₁ + b₂) × h

Triangle Area = (1/2) × base × height
Circle Area = π × r²

▸ Quick Example

A triangle has base 8 and height 5. Find its area.
A = (1/2)(8)(5) = 20

Answer: 20

07

Pythagorean Theorem & Right Triangles

The Pythagorean Theorem and its common triples appear in nearly every AMC 8 geometry section.

▸ Key Facts to Memorize

◆a² + b² = c² where c is the hypotenuse

◆Pythagorean triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25)

◆Multiples of triples also work: (6,8,10), (9,12,15)

a² + b² = c²
Common triples: 3-4-5, 5-12-13, 8-15-17

▸ Quick Example

Legs 5 and 12 — find the hypotenuse.
c² = 25 + 144 = 169, c = 13

Answer: 13

08

Counting & Probability

Systematic listing, multiplication principle, and basic probability are essential AMC 8 tools.

▸ Key Facts to Memorize

◆Multiplication Principle: if event A has m outcomes and B has n, there are m×n total

◆P(event) = favorable outcomes / total equally likely outcomes

◆P(A or B) = P(A) + P(B) − P(A and B)

◆n! = n × (n−1) × ... × 1

P(event) = (favorable outcomes) / (total outcomes)
Arrangements of n distinct items = n!

▸ Quick Example

A bag has 3 red and 2 blue balls. P(picking red)?
P = 3/5

Answer: 3/5

09

Statistics: Mean, Median, Mode, Range

AMC 8 frequently tests which measure of center is affected by outliers and data changes.

▸ Key Facts to Memorize

◆Mean = sum of values / count

◆Median = middle value when sorted (average of two middles if even count)

◆Mode = most frequent value

◆Range = max − min

Mean = (sum of all values) / (number of values)

▸ Quick Example

Data: 4, 7, 2, 9, 3. Find the median.
Sorted: 2, 3, 4, 7, 9. Middle = 4

Answer: 4

10

Patterns, Sequences & Logic

AMC 8 uses arithmetic sequences, geometric sequences, and cyclical patterns (last digits, remainders).

▸ Key Facts to Memorize

◆Arithmetic sequence: a_n = a₁ + (n−1)d; Sum = n(a₁ + aₙ)/2

◆Geometric sequence: a_n = a₁ × r^(n−1)

◆Last digit of powers of 2 cycles: 2,4,8,6 (period 4)

Arithmetic: aₙ = a₁ + (n−1)d
Sum of first n terms: S = n(a₁ + aₙ) / 2

▸ Quick Example

Find the 10th term: 3, 7, 11, 15, ...
a₁=3, d=4. a₁₀ = 3 + 9×4 = 3 + 36 = 39

Answer: 39