Topic 01 Algebra & Equations

Core Concepts

Solving linear & quadratic equations; Vieta's Formulas
Systems of equations (substitution, elimination)
Absolute value equations & inequalities
Arithmetic & geometric sequences/series

Must Memorize

Quadratic: ax² + bx + c = 0 ⇒ x = (−b ± √(b²−4ac)) / 2a
Vieta: x₁+x₂ = −b/a x₁·x₂ = c/a
Sum of AP: Sₙ = n(a₁+aₙ)/2 GP: Sₙ = a(1−rⁿ)/(1−r)

Quick Example

If x² − 5x + 6 = 0, find x₁·x₂.

By Vieta: product = c/a = 6/1 = 6

Topic 02 Number Theory

Core Concepts

Divisibility rules, GCD, LCM
Prime factorization & number of divisors
Modular arithmetic & remainders
Units digits, last digit patterns

Must Memorize

If n = p₁ᵃ¹ · p₂ᵃ² · …, then # divisors = (a₁+1)(a₂+1)…
GCD(a,b) · LCM(a,b) = a · b
a ≡ b (mod m) means m | (a−b)

Quick Example

How many divisors does 72 = 2³·3² have?

(3+1)(2+1) = 12

Topic 03 Plane Geometry

Core Concepts

Triangle properties: area, Pythagorean theorem, special triangles
Circle theorems: inscribed angle, arc length, area
Similar triangles, proportionality
Coordinate geometry: distance, slope, midpoint

Must Memorize

Special triangles: 30-60-90 (1 : √3 : 2) 45-45-90 (1 : 1 : √2)
Circle area = πr² Circumference = 2πr
Heron: A = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2

Quick Example

A right triangle has legs 3 and 4. Find hypotenuse.

√(9+16) = √25 = 5

Topic 04 Counting & Combinatorics

Core Concepts

Permutations P(n,r) and Combinations C(n,r)
Multiplication & Addition Principles
Complementary counting
Pascal's Triangle & Binomial Theorem

Must Memorize

P(n,r) = n!/(n−r)! C(n,r) = n! / (r!(n−r)!)
C(n,r) = C(n, n−r) C(n,0) = C(n,n) = 1
(a+b)ⁿ = ∑ C(n,k) aⁿ⁻ᵏ bᵏ

Quick Example

How many ways to choose 3 from 7?

C(7,3) = 35 35

Topic 05 Probability

Core Concepts

Basic probability: P(A) = favorable / total
Independent events: P(A and B) = P(A)·P(B)
Mutually exclusive: P(A or B) = P(A)+P(B)
Complementary: P(not A) = 1 − P(A)

Must Memorize

0 ≤ P(A) ≤ 1 P(sample space) = 1
P(A or B) = P(A)+P(B)−P(A and B)
Complementary: use 1−P(event) for “at least one”

Quick Example

P(at least one head in 2 flips) = 1 − P(TT) = 1 − 1/4 = 3/4

Topic 06 Functions & Graphs

Core Concepts

Domain, range, inverse functions
Transformations: shifts, reflections, stretches
Floor/ceiling functions ⌊x⌋, ⌈x⌉
Piecewise functions

Must Memorize

f(f⁻¹(x)) = x (f∘g)(x) = f(g(x))
⌊x⌋ = greatest integer ≤ x
Vertex of ax²+bx+c: x = −b/(2a)

Quick Example

If f(x) = 2x+1, find f⁻¹(5). Set 2x+1=5, x=2. Answer: 2

Topic 07 Word Problems & Rates

Core Concepts

Distance = Rate × Time
Work problems: combined rate = 1/a + 1/b
Mixture and percentage problems
Age problems, ratio problems

Must Memorize

d = r · t Work W = R · T
Combined work rate: 1/T = 1/a + 1/b
Average speed = Total distance / Total time

Quick Example

A can do a job in 4 hrs, B in 6 hrs. Together: 1/T = 1/4 + 1/6 = 5/12, so T = 12/5 hrs

Topic 08 Logic & Patterns

Core Concepts

Pattern recognition in sequences
Pigeonhole principle
Proof by contradiction; casework
Extremal principle

Must Memorize

Pigeonhole: n+1 items into n holes ⇒ some hole has ≥2
Casework: systematically split into non-overlapping cases
Invariant: find a quantity that never changes

Quick Example

13 people ⇒ at least 2 share a birth month. (12 months, pigeonhole) ✓

Problem 01 Algebra Easy

If the roots of x² − 7x + k = 0 are consecutive integers, what is the value of k?

Let the roots be n and n+1. By Vieta: n+(n+1) = 7, so 2n+1 = 7, n = 3. The roots are 3 and 4. Product k = 3×4 = 12.

Problem 02 Algebra — Sequences Medium

The sum of the first 50 positive odd integers is:

The first 50 odd integers are 1, 3, 5, …, 99. Key identity: the sum of first n odd integers = n². So the answer is 50² = 2500.

Problem 03 Number Theory Easy

How many positive divisors does 360 have?
(360 = 2³ · 3² · 5)

360 = 2³ · 3² · 5¹. Number of divisors = (3+1)(2+1)(1+1) = 4 · 3 · 2 = 24.

Problem 04 Number Theory — Modular Arithmetic Medium

What is the units digit of 7⁹¹?

Units digits of powers of 7 cycle with period 4: 7¹→7, 7²→9, 7³→3, 7⁴→1, then repeat. Since 91 = 4×22 + 3, the units digit equals that of 7³, which is 3. Wait — let me recount: 91 mod 4 = 3, so units digit of 7⁹¹ = units digit of 7³ = 3.

Correction to answer key: 91 ÷ 4 = 22 remainder 3. Cycle position 3 gives units digit 3. The correct answer is (C) 3.

Problem 05 Geometry — Triangles Easy

In a 30-60-90 triangle, the hypotenuse has length 10. What is the area of the triangle?

In a 30-60-90 triangle with hypotenuse 10: short leg = 5, long leg = 5√3. Area = (1/2)(5)(5√3) = 25√3/2.

Problem 06 Geometry — Circles Medium

A circle has area 49π. A chord is drawn at distance 3 from the center. What is the length of the chord?

Area = πr² = 49π ⇒ r = 7. The perpendicular from center to chord bisects it. Half-chord = √(r² − d²) = √(49 − 9) = √40 = 2√10. Full chord = 2 × 2√10 = 4√10.

Correct answer: (A) 4√10.

Problem 07 Combinatorics Easy

How many 3-digit numbers can be formed using the digits {1, 2, 3, 4, 5} with no digit repeated?

P(5,3) = 5 × 4 × 3 = 60. (5 choices for hundreds, 4 for tens, 3 for units.)

Problem 08 Combinatorics — Complementary Counting Medium

How many ways can 4 people be arranged in a row if two specific people (Alex and Bob) must NOT be adjacent?

Total arrangements = 4! = 24. Arrangements where Alex & Bob ARE adjacent: treat them as a block → 3! × 2! = 6 × 2 = 12. Non-adjacent = 24 − 12 = 12.

Problem 09 Probability Easy

A bag contains 4 red, 3 blue, and 5 green marbles. One marble is drawn at random. What is the probability it is NOT red?

Total = 12. P(red) = 4/12 = 1/3. P(not red) = 1 − 1/3 = 2/3.

Problem 10 Probability — Dice Medium

Two fair six-sided dice are rolled. What is the probability that the sum equals 8?

Pairs summing to 8: (2,6),(3,5),(4,4),(5,3),(6,2) → 5 pairs. Total outcomes = 36. P = 5/36.

Problem 11 Functions Medium

If f(x) = x² − 2x + 3, what is the minimum value of f(x)?

Vertex at x = −b/(2a) = −(−2)/(2) = 1. f(1) = 1 − 2 + 3 = 2. This is the minimum since a = 1 > 0.

Problem 12 Functions — Composition Hard

If f(x) = 2x + 1 and g(x) = x², compute f(g(f(1))).

Step 1: f(1) = 2(1)+1 = 3.
Step 2: g(f(1)) = g(3) = 3² = 9.
Step 3: f(g(f(1))) = f(9) = 2(9)+1 = 19.

Problem 13 Word Problems — Rates Medium

Alice drives from city A to city B at 60 mph, then returns at 40 mph. What is her average speed for the entire trip?

Harmonic mean for equal distances: Avg speed = 2×60×40 / (60+40) = 4800/100 = 48 mph. Never just average the speeds — use total distance / total time.

Problem 14 Word Problems — Work Hard

Pipe A fills a tank in 6 hours. Pipe B drains it in 9 hours. If both are open simultaneously starting from empty, how many hours to fill the tank?

Net rate = 1/6 − 1/9 = 3/18 − 2/18 = 1/18 tanks/hour. Time = 18/1 = 18 hours.

Problem 15 Number Theory — GCD/LCM Hard

The GCD of two numbers is 12 and their LCM is 180. If one number is 36, what is the other?

GCD × LCM = a × b. So 12 × 180 = 36 × b. b = 2160 / 36 = 60.
Check: GCD(36,60) = 12 ✓ and LCM(36,60) = 180 ✓.

Problem 16 Geometry — Similar Triangles Hard

In triangle ABC, D is on BC such that AD bisects angle A. If AB = 6, AC = 9, and BC = 10, find BD.

Angle Bisector Theorem: BD/DC = AB/AC = 6/9 = 2/3. Since BD + DC = 10, we have BD = (2/5) × 10 = 4 and DC = 6.

Problem 17 Combinatorics — Pascal's Triangle Hard

What is the coefficient of x³ in the expansion of (2x + 3)⁵?

Term with x³ in (2x+3)⁵: use k=3. C(5,3) · (2x)³ · 3² = 10 · 8x³ · 9 = 720x³. Coefficient = 720.

Problem 18 Algebra — Systems Hard

If x + y = 7 and xy = 10, find x² + y².

Identity: x²+y² = (x+y)² − 2xy = 7² − 2(10) = 49 − 20 = 29.

Problem 19 Logic & Patterns Medium

A sequence is defined by a₁ = 1, a₂ = 1, and aₙ = aₙ⁻¹ + aₙ⁻² for n ≥ 3 (Fibonacci). What is a₃₀?

The Fibonacci sequence: 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657,46368,75025,121393,196418,317811,514229,832040.
The 30th Fibonacci number is 832040.

Problem 20 Mixed — Geometry & Algebra Hard

A square has side length s. Its diagonal is also the diameter of a circle. What is the ratio of the circle's area to the square's area?

Square diagonal = s√2. Circle diameter = s√2, so radius = s√2/2.
Circle area = π(s√2/2)² = π · s²·2/4 = πs²/2.
Square area = s².
Ratio = (πs²/2) / s² = π/2.

AMC 10 Master Workbook — 20 Essential Problems

Topics: Algebra • Number Theory • Geometry • Combinatorics • Probability • Functions • Rates • Logic

Time limit: 75 minutes • 5 answer choices per problem

If the roots of x² − 7x + k = 0 are consecutive integers, what is k?

(A) 10(B) 11(C) 12(D) 13(E) 14

The sum of the first 50 positive odd integers is:

How many positive divisors does 360 have? (360 = 2³·3²·5)

(A) 18(B) 20(C) 24(D) 30(E) 36

What is the units digit of 7⁹¹?

(A) 1(B) 7(C) 3(D) 9(E) 4

In a 30-60-90 triangle, the hypotenuse has length 10. What is the area?

(A) 10(B) 20(C) 25√3/2(D) 5√3(E) 50

A circle has area 49π. A chord is drawn at distance 3 from the center. What is the length of the chord?

(A) 4√10(B) 12(C) 10√2(D) 8√10(E) 2√40

How many 3-digit numbers can be formed from {1,2,3,4,5} with no digit repeated?

(A) 10(B) 30(C) 60(D) 120(E) 125

How many ways can 4 people be arranged in a row if Alex and Bob must NOT be adjacent?

(A) 8(B) 12(C) 16(D) 18(E) 20

A bag has 4 red, 3 blue, 5 green marbles. One is drawn. P(not red) = ?

(A) 1/3(B) 5/12(C) 1/2(D) 2/3(E) 3/4

10.

Two fair dice are rolled. P(sum = 8) = ?

(A) 1/6(B) 1/9(C) 5/36(D) 7/36(E) 1/4

11.

If f(x) = x² − 2x + 3, what is the minimum value of f(x)?

(A) 1(B) 2(C) 3(D) 0(E) −1

12.

If f(x)=2x+1 and g(x)=x², compute f(g(f(1))).

(A) 15(B) 17(C) 18(D) 19(E) 21

13.

Alice drives from A to B at 60 mph and returns at 40 mph. Average speed for entire trip?

(A) 45(B) 48(C) 50(D) 52(E) 54

14.

Pipe A fills a tank in 6 hours. Pipe B drains it in 9 hours. Both open from empty — hours to fill?

(A) 15(B) 16(C) 18(D) 20(E) 27

15.

GCD of two numbers is 12, LCM is 180. One number is 36. Find the other.

(A) 48(B) 60(C) 72(D) 84(E) 90

16.

In triangle ABC, AD bisects angle A with D on BC. AB=6, AC=9, BC=10. Find BD.

(A) 4(B) 5(C) 6(D) 3(E) 2

17.

What is the coefficient of x³ in (2x+3)⁵?

(A) 480(B) 600(C) 720(D) 1080(E) 1440

18.

If x+y=7 and xy=10, find x²+y².

(A) 27(B) 29(C) 31(D) 39(E) 49

19.

In the Fibonacci sequence (a₁=1, a₂=1, aₙ=aₙ−¹+aₙ−²), what is a₃₀?

(A) 377(B) 499(C) 514229(D) 832040(E) 1346269

20.

A square has side s. Its diagonal is the diameter of a circle. Ratio of circle area to square area?

(A) π/4(B) π/3(C) π/2(D) π(E) 2π/3

Answer Key & Solutions

1. (C) 12

Let roots be n, n+1. Sum: 2n+1=7 so n=3. Product: 3×4=12.

2. (E) 2500

Sum of first n odd integers = n². For n=50: 50²=2500.

3. (C) 24

360=2³·3²·5¹. Divisors=(3+1)(2+1)(1+1)=24.

4. (C) 3

Powers of 7 cycle (period 4): 7,9,3,1. 91 mod 4 = 3. Units digit = 3.

5. (C) 25√3/2

Hyp=10, short leg=5, long leg=5√3. Area=(1/2)(5)(5√3)=25√3/2.

6. (A) 4√10

r=7. Half-chord=√(49−9)=√40=2√10. Full chord=4√10.

7. (C) 60

P(5,3)=5×4×3=60.

8. (B) 12

Total=24. Adjacent cases=3!×2!=12. Non-adjacent=24−12=12.

9. (D) 2/3

P(not red)=1−4/12=2/3.

10. (C) 5/36

Pairs summing to 8: (2,6),(3,5),(4,4),(5,3),(6,2)=5. P=5/36.

11. (B) 2

Vertex x=1. f(1)=1−2+3=2.

12. (D) 19

f(1)=3, g(3)=9, f(9)=19.

13. (B) 48 mph

Harmonic mean: 2(60)(40)/(60+40)=4800/100=48.

14. (C) 18

Net rate=1/6−1/9=1/18. Time=18 hrs.

15. (B) 60

12×180=36×b. b=2160/36=60.

16. (A) 4

Angle Bisector: BD/DC=6/9=2/3. BD=(2/5)×10=4.

17. (C) 720

C(5,3)(2)³(3)²=10×8×9=720.

18. (B) 29

x²+y²=(x+y)²−2xy=49−20=29.

19. (D) 832040

The 30th Fibonacci number is 832040.

20. (C) π/2

Diagonal=s√2, radius=s√2/2. Circle area=πs²/2. Ratio=π/2.

20 Essential Problems
All Core Topics

Core Concepts

Core Concepts

Core Concepts

Core Concepts

Core Concepts

Core Concepts

Core Concepts

Core Concepts

Practice Problems

Final Score

AMC 10 Master Workbook — 20 Essential Problems

Answer Key & Solutions

20 Essential ProblemsAll Core Topics

Core Concepts

Core Concepts

Core Concepts

Core Concepts

Core Concepts

Core Concepts

Core Concepts

Core Concepts

Practice Problems

Final Score

AMC 10 Master Workbook — 20 Essential Problems

Answer Key & Solutions

20 Essential Problems
All Core Topics