AMC 10 Master Practice | 20 Essential Problems

1

Algebra — Equations & Expressions

▼

Quadratic Formulax = (−b ± √(b²−4ac)) / (2a)

Vieta's Formulas (ax²+bx+c=0)Sum of roots = −b/a Product of roots = c/a

Key Identitiesa²−b² = (a+b)(a−b) (a+b)² = a²+2ab+b² (a−b)² = a²−2ab+b²

🧠 Memorize • Quadratic with sum S and product P: x²−Sx+P = 0
• AM–GM: (a+b)/2 ≥ √(ab) for non-negative a, b
• (x + 1/x)² = x² + 2 + 1/x² → x²+1/x² = (x+1/x)²−2

📒 Example Roots of x²−5x+6=0 are r and s. Find r²+s².

r+s=5, rs=6 → r²+s² = (r+s)²−2rs = 25−12 = 13

2

Geometry — Triangles & Circles

▼

Area FormulasTriangle: A = ½bh Circle: A = πr² Sector: A = (θ/360)·πr² Circumference = 2πr

Pythagorean Theorema²+b² = c² Common triples: 3-4-5, 5-12-13, 8-15-17 30-60-90: sides 1:√3:2 45-45-90: sides 1:1:√2

Circle Chord TheoremIf chord is distance d from center, half-length = √(r²−d²)

🧠 Memorize • Inscribed angle = ½ × central angle (same arc)
• Similar triangles: ratio of sides k → ratio of areas k²
• Regular hexagon with side s: area = (3√3/2)s² = 6×(√3/4)s²

📒 Example Right triangle with legs 6 and 8. Find area and hypotenuse.

Hypotenuse = 10 (3-4-5 scaled ×2). Area = ½(6)(8) = 24

3

Number Theory — Divisibility & Primes

▼

Number of DivisorsIf n = p₁ᵃ·p₂ᵇ·p₃ᶜ…, then τ(n) = (a+1)(b+1)(c+1)…

Sum of Divisorsσ(n) = (1+p₁+p₁²+…+p₁ᵃ)(1+p₂+…+p₂ᵇ)…

GCD & LCMGCD(a,b) × LCM(a,b) = a × b

🧠 Memorize • Powers of 7 cycle in units digits: 7, 9, 3, 1 (period 4)
• Powers of 2 mod 7: 2, 4, 1 (period 3)
• For n mod k: find the cycle, compute exponent mod period
• Divisibility by 3: digit sum divisible by 3

📒 Example How many positive divisors does 360 have?

360 = 2³·3²·5¹ → τ(360) = (3+1)(2+1)(1+1) = 4·3·2 = 24

4

Combinatorics — Counting & Arrangements

▼

Permutations & CombinationsP(n,r) = n!/(n−r)! C(n,r) = n!/[r!(n−r)!]

Inclusion-Exclusion|A∪B| = |A|+|B|−|A∩B| |A∪B∪C| = |A|+|B|+|C|−|A∩B|−|A∩C|−|B∩C|+|A∩B∩C|

🧠 Memorize • Circular permutation of n: (n−1)!
• Choose-then-arrange trick: multiply C(n,r) × r! to get P(n,r)
• Increasing-digit numbers from {1,…,9}: each 4-subset gives exactly one

📒 Example In how many ways can 3 boys and 3 girls sit with no two boys adjacent?

Arrange girls: 3!=6. Gaps created: 4. Choose 3 gaps: C(4,3)=4. Arrange boys: 3!=6. Total = 6×4×6 = 144

5

Probability & Expected Value

▼

Basic ProbabilityP(A) = (favorable outcomes) / (total outcomes) P(A∪B) = P(A)+P(B)−P(A∩B)

Complementary ProbabilityP(A) = 1 − P(A⊃c⊃) ← Often the fastest method!

Expected ValueE[X] = ∑ xᵢ·P(xᵢ)

🧠 Memorize • Without replacement: probabilities change each draw
• P(both same color) = P(both A) + P(both B)
• Product is even ⇔ NOT (both odd). P(both odd) = (1/2)² = 1/4
• Geometric probability = favorable area / total area

📒 Example Fair coin flipped 3 times. P(exactly 2 heads)?

C(3,2)·(1/2)²·(1/2) = 3·(1/8) = 3/8

6

Sequences, Series & Logarithms

▼

Arithmetic Sequenceaₙ = a₁+(n−1)d Sₙ = n(a₁+aₙ)/2

Geometric Sequenceaₙ = a₁·rⁿ₋¹ S∞ = a₁/(1−r) for |r|<1

Logarithm Lawslogᵃ(xy) = logᵃx+logᵃy logᵃ(xⁿ) = n·logᵃx Change of base: logᵃb = log(b)/log(a)

🧠 Memorize • aₙ = Sₙ − Sₙ⁻¹ (nth term from partial sum formula)
• Sum of first n integers: n(n+1)/2
• log₃(x) + log₁₀(x) = (3/2)log₁₀(x) after converting
• Geometric: a₄/a₂ = r² (ratio of terms 2 apart = r²)

📒 Example Infinite geometric series: 1 + 1/2 + 1/4 + …. Find the sum.

a₁=1, r=1/2. S = 1/(1−1/2) = 1/(1/2) = 2

20 Essential Problems

Concept Review

20 AMC-Style Problems

Answer Key & Full Solutions