Official Prep Book
AMC 8
Master Problem Set
All Topics · 20 Questions · Full Solutions
20
Problems
10
Topics
40min
Timer

Core Concepts & Formulas

Review before attempting the problems

1
Arithmetic & Number Theory
GCD (Greatest Common Divisor): largest number dividing both. LCM × GCD = a × b
Divisibility rules: ÷2 (even), ÷3 (digit sum ÷3), ÷4 (last 2 digits ÷4), ÷5 (ends 0,5), ÷9 (digit sum ÷9)
Prime factorization: express as product of primes. Number of divisors = (e₁+1)(e₂+1)…
LCM(a,b) = (a × b) / GCD(a,b) Sum of 1 to n = n(n+1)/2 Sum of odd integers 1 to (2n-1) = n²
Example
How many positive divisors does 36 have?
36 = 2² × 3² → divisors = (2+1)(2+1) = 9
Answer: 9 divisors (1,2,3,4,6,9,12,18,36)
2
Fractions, Decimals & Percentages
Percent change = (new − old) / old × 100%
Two successive percent changes of a% then b%: overall = a + b + ab/100
x% of y = y% of x = xy/100 Discount: new price = original × (1 − rate) Markup: new price = original × (1 + rate)
Example
A price increases 20% then decreases 20%. Net change?
Overall = 20 + (−20) + (20×−20)/100 = 0 − 4 = −4%
Answer: 4% decrease (not 0%!)
3
Algebra & Equations
Linear equations: isolate the variable using inverse operations
Systems of equations: substitution or elimination method
Average (Mean) = Sum / Count Weighted average: (w₁a₁ + w₂a₂) / (w₁ + w₂) Work rate: if A finishes in a days, rate = 1/a per day
Example
If 3x + 7 = 22, find x.
3x = 15, so x = 5
Answer: x = 5
4
Geometry — Area & Perimeter
Rectangle: Area = l×w, Perimeter = 2(l+w) Triangle: Area = ½×b×h Circle: Area = πr², Circumference = 2πr Trapezoid: Area = ½×(b₁+b₂)×h
Pythagorean theorem: a² + b² = c² (right triangle)
Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25
Example
Area of a triangle with base 10 and height 6?
Area = ½ × 10 × 6 = 30
Answer: 30 square units
5
Counting & Probability
Permutations: nPr = n! / (n−r)! Combinations: nCr = n! / [r!(n−r)!] Probability = favorable outcomes / total outcomes Complement: P(event) = 1 − P(not event)
Counting Principle: if event A has m ways and B has n ways, together = m × n ways
P(A or B) = P(A) + P(B) − P(A and B)
Example
Choose 2 from {A,B,C,D}: how many ways? C(4,2) = 4!/(2!×2!) = 6
Answer: 6 combinations
6
Statistics & Data
Mean = sum of values / number of values Median = middle value when sorted Mode = most frequent value Range = max − min
If a value is added to a set, recalculate mean using the new total sum and count
Example
Median of {3, 7, 1, 9, 5}? Sort: 1,3,5,7,9 → middle = 5
Answer: Median = 5
Question 1 of 20
AMC 8 · Master Set
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