AMC 10 · Master Series

AMC 10
Practice Test

20 Problems · All Major Topics · Competition Style

20
Problems
75
Minutes
5
Choices
Number Theory Algebra Geometry Combinatorics Probability Functions Sequences Inequalities Logarithms Word Problems

Concept Review

Key formulas and memorization points for all AMC 10 topics

1
Number Theory — Divisibility & GCD/LCM
GCD(a,b) × LCM(a,b) = a × b
a | b means "a divides b"
Number of divisors of n = p₁^a₁·p₂^a₂·… is (a₁+1)(a₂+1)…

📌 Memorize

  • Divisibility by 3: digit sum divisible by 3
  • Divisibility by 9: digit sum divisible by 9
  • Divisibility by 11: alternating digit sum divisible by 11
  • Perfect squares have odd number of divisors
ExampleFind the number of divisors of 360 = 2³·3²·5¹
→ (3+1)(2+1)(1+1) = 24 divisors ✓
2
Algebra — Quadratics & Vieta's Formulas
ax² + bx + c = 0
Sum of roots: r₁ + r₂ = -b/a
Product of roots: r₁ · r₂ = c/a
Discriminant: Δ = b² - 4ac

📌 Memorize

  • Δ > 0: two real roots; Δ = 0: one real root; Δ < 0: no real roots
  • (a+b)² = a²+2ab+b² ; (a−b)² = a²−2ab+b²
  • a²−b² = (a+b)(a−b)
  • a³+b³ = (a+b)(a²−ab+b²)
ExampleIf roots of x²−5x+6=0 are r,s, find r²+s².
→ r+s=5, rs=6, so r²+s²=(r+s)²−2rs=25−12=13 ✓
3
Geometry — Triangles & Circles
Triangle area = ½·b·h = ½·a·b·sinC
Heron: s=(a+b+c)/2, Area=√(s(s−a)(s−b)(s−c))
Circle: A=πr², C=2πr
Sector area: (θ/360°)·πr²

📌 Memorize

  • 30-60-90: sides 1 : √3 : 2
  • 45-45-90: sides 1 : 1 : √2
  • Pythagorean triples: (3,4,5),(5,12,13),(8,15,17)
  • Inscribed angle = half central angle
  • Power of a point: PA·PB = PC·PD
ExampleA right triangle has legs 5 and 12. Hypotenuse = √(25+144) = 13 ✓
4
Combinatorics — Counting Principles
Permutations: P(n,r) = n!/(n−r)!
Combinations: C(n,r) = n!/[r!(n−r)!]
Stars & Bars: C(n+k−1, k) ways to place k identical items in n bins

📌 Memorize

  • Addition Principle: OR → add cases
  • Multiplication Principle: AND → multiply cases
  • Complementary Counting: Total − Unwanted
  • Pascal's Triangle: C(n,r)+C(n,r+1)=C(n+1,r+1)
ExampleHow many ways to choose 3 from 7? C(7,3)=35 ✓
5
Probability & Expected Value
P(A) = favorable / total
P(A and B) = P(A)·P(B) [independent]
P(A or B) = P(A)+P(B)−P(A and B)
E[X] = Σ xᵢ·P(xᵢ)

📌 Memorize

  • Complementary probability: P(at least one) = 1−P(none)
  • Geometric probability: favorable length/area over total
  • Conditional: P(A|B) = P(A∩B)/P(B)
ExampleProbability of rolling at least one 6 in 2 rolls = 1−(5/6)²=11/36 ✓
6
Sequences, Series & Logarithms
Arithmetic: aₙ = a₁+(n−1)d, Sₙ = n(a₁+aₙ)/2
Geometric: aₙ = a₁·rⁿ⁻¹, S∞ = a₁/(1−r) for |r|<1
log_b(xy)=log_b(x)+log_b(y)
log_b(xⁿ)=n·log_b(x)

📌 Memorize

  • Change of base: log_b(x) = ln(x)/ln(b)
  • log_b(b) = 1 ; log_b(1) = 0
  • Infinite geometric series converges only when |r| < 1
Examplelog₂(8)+log₂(4) = 3+2 = 5 ; also = log₂(32) = 5 ✓
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Time Left
75:00
Problem 1 of 20