AMC 8 Master Prep

The Essential 20
Problem Collection

All five topics · Competition style · Full solutions

📐 Geometry 🔢 Number Theory 📊 Statistics 🎲 Combinatorics ∑ Algebra

Core Concepts & Formulas

Review every key formula and memorization item before tackling the problems.

🔢

Number Theory

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Essential Formulas

Divisibility — sum of digits
Div by 3 or 9: digit sum divisible by 3 or 9
Number of divisors
n = p₁ᵃ·p₂ᵇ … → τ(n) = (a+1)(b+1)…
GCD × LCM identity
GCD(a,b) × LCM(a,b) = a × b
Sum of first n positive integers
1+2+…+n = n(n+1)/2
Units digit patterns (powers)
2: 2,4,8,6 (cycle 4) | 3: 3,9,7,1 | 7: 7,9,3,1

Must Memorize

  • Primes to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
  • Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169
  • Perfect cubes: 1, 8, 27, 64, 125, 216
  • A perfect square has an ODD number of total divisors
  • Div by 11: alternate digit difference divisible by 11
  • Consecutive integer products are always divisible by 2; three consecutive by 6
∑

Algebra & Arithmetic

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Essential Formulas

Percent change
% change = (new − old)/old × 100
Distance / Rate / Time
d = r × t
Arithmetic sequence sum
S = n × (first + last) / 2
Average (mean)
Mean = (sum of values) / (count)
Work rate
Combined rate = 1/a + 1/b (jobs/unit time)

Must Memorize

  • If average of n numbers is A, their sum = n × A
  • Consecutive integers: n, n+1, n+2 … ; their average is the middle term
  • 30-60-90 side ratio: 1 : √3 : 2; 45-45-90: 1 : 1 : √2
  • Simple interest: I = Prt; Compound: A = P(1 + r)ⁿ
  • Always check: does the answer make sense in context?
📐

Geometry

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Essential Formulas

Circle: Area & Circumference
A = πr² | C = 2πr = πd
Triangle: Area
A = ½ × base × height
Pythagorean Theorem
a² + b² = c² (c = hypotenuse)
Trapezoid Area
A = ½(b₁ + b₂) × h
Sum of interior angles (polygon)
(n − 2) × 180°
Volume: Rectangular Prism & Cylinder
V = l×w×h | V = πr²h

Must Memorize

  • Pythagorean triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25)
  • Vertical angles are equal; supplementary angles sum to 180°
  • Similar triangles: corresponding sides are proportional
  • Arc length = (θ/360°) × 2πr; Sector area = (θ/360°) × πr²
  • Diagonal of a square with side s: d = s√2
🎲

Combinatorics & Probability

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Essential Formulas

Fundamental Counting Principle
Total = a × b × c × …
Permutations (ordered)
P(n,r) = n!/(n−r)!
Combinations (unordered)
C(n,r) = n!/[r!(n−r)!]
Basic Probability
P(E) = favorable outcomes / total outcomes
Complement Rule
P(not E) = 1 − P(E)

Must Memorize

  • n! = n × (n−1) × … × 2 × 1 (0! = 1)
  • Arrangements of n objects with repeats: n! / (r₁! × r₂! × …)
  • Pascal's Triangle row n gives C(n, 0) through C(n, n)
  • P(A and B) = P(A) × P(B) only when A, B are independent
  • Handshake problem: n people → C(n,2) = n(n−1)/2 handshakes
📊

Statistics & Data

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Essential Formulas

Mean (average)
Mean = Σxᵢ / n
Median
Middle value when data is sorted; average of two middles if n is even
Mode
Most frequently occurring value
Range
Range = maximum − minimum
Weighted Average
W.A. = Σ(value × weight) / Σweight

Must Memorize

  • Adding a constant k to every value changes mean by k, not standard deviation
  • Multiplying every value by k multiplies both mean and range by k
  • Outliers affect mean more than median — median is more "resistant"
  • For n data points, if n is odd, median is the ((n+1)/2)th value
  • Read bar, line, and pie charts carefully — watch scale and labels

Worked Examples

One solved problem per topic — study the approach, then try the practice set.

1

Number Theory — Divisors & Prime Factorization

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How many positive integers less than 100 are divisible by both 4 and 6?
Solution
1Find LCM(4, 6). Since 4 = 2² and 6 = 2 × 3, we get LCM = 2² × 3 = 12.
2Count multiples of 12 that are less than 100: 12, 24, 36, 48, 60, 72, 84, 96.
3That gives ⌊99/12⌋ = 8 multiples.
Answer8
2

Algebra — Average & Sum

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The average of five numbers is 18. When a sixth number is added, the average becomes 20. What is the sixth number?
Solution
1Sum of first five numbers = 5 × 18 = 90.
2Sum of all six numbers = 6 × 20 = 120.
3Sixth number = 120 − 90 = 30.
Answer30
3

Geometry — Area & Pythagorean Theorem

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A right triangle has legs of length 9 and 12. What is the area of the triangle?
Solution
1For a right triangle, the two legs serve as base and height.
2Area = ½ × 9 × 12 = ½ × 108 = 54.
3(Bonus: hypotenuse = √(81 + 144) = √225 = 15, confirming a 3-4-5 triple scaled by 3.)
Answer54
4

Combinatorics — Counting Arrangements

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In how many ways can 4 different books be arranged on a shelf?
Solution
1Position 1: 4 choices. Position 2: 3 choices. Position 3: 2 choices. Position 4: 1 choice.
2Total = 4! = 4 × 3 × 2 × 1 = 24.
Answer24
5

Statistics — Median of a Data Set

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Find the median of the data set: {7, 3, 11, 5, 9, 1, 15}.
Solution
1Sort in ascending order: 1, 3, 5, 7, 9, 11, 15.
2There are 7 values (odd), so median = the 4th value = 7.
Answer7
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