Competition Math

AMC 8 Master Practice

20 Essential Problems · All Core Topics · Competition Style

20
Problems
10
Topics
40
Min
⏱ 40:00
📐
Concept Review
Key Formulas & Rules to Memorize
🔢 Number Theory
Divisibility rules, LCM, GCF, prime factorization. A number is prime if it has exactly 2 factors.
LCM × GCF = a × b
🔀 Fractions & Percents
Percent change = (new − old)/old × 100%. Convert fractions to decimals by dividing.
% = (part/whole) × 100
📏 Geometry — 2D
Area of triangle = ½bh. Circle area = πr². Rectangle area = lw. Sum of angles in triangle = 180°.
A = πr², C = 2πr
📊 Statistics
Mean = sum ÷ count. Median = middle value (sorted). Mode = most frequent. Range = max − min.
Mean = Σx / n
🎲 Probability
P(event) = favorable outcomes / total outcomes. Values always between 0 and 1.
P = favorable / total
📐 Algebra
Solve by isolating variables. Exponent rule: a^m × a^n = a^(m+n). Distribute before combining.
a(b+c) = ab + ac
⏱ Rate & Ratio
Distance = Rate × Time. Work problems: add rates. If A does job in a hours, rate = 1/a.
d = r × t
🧮 Counting
Fundamental counting principle: multiply choices. Combinations: C(n,r) = n! / (r!(n−r)!)
C(n,2) = n(n−1)/2
📦 3D Geometry
Box volume = l×w×h. Cylinder volume = πr²h. Surface area of box = 2(lw + lh + wh).
V_box = lwh
🔣 Patterns & Sequences
Arithmetic: each term adds constant d. nth term = a + (n−1)d. Look for remainders in cycles.
a_n = a₁ + (n−1)d
💡
Worked Examples
Study these before solving
Example 1 · Number Theory
What is the LCM of 12 and 18?
12 = 2²×3, 18 = 2×3². LCM = 2²×3² = 36.
Example 2 · Geometry
A right triangle has legs 3 and 4. What is its hypotenuse?
By Pythagorean theorem: 3²+4² = 9+16 = 25, so hypotenuse = 5.
Example 3 · Probability
A bag has 3 red and 5 blue marbles. What is P(red)?
Total = 8. P(red) = 3/8 = 0.375.
Example 4 · Percent
A shirt costs $40 and is discounted 25%. What is the sale price?
Discount = 0.25×40 = $10. Sale price = 40−10 = $30.
Practice Problems
Answer all 20 questions · Choose the best answer
Problem 1
Number Theory — Divisibility
How many positive integers less than 50 are divisible by both 3 and 4?
Problem 2
Fractions — Ordering
Which of the following fractions is largest?
Problem 3
Percentages — Percent Change
A school's enrollment increased from 800 students to 1,000 students. By what percent did enrollment increase?
Problem 4
Ratios & Proportions
A recipe uses flour and sugar in a ratio of 5 : 2. If a baker uses 15 cups of flour, how many cups of sugar are needed?
Problem 5
Geometry — Area
A rectangle has a perimeter of 36 cm and a length of 11 cm. What is its area in square centimeters?
Problem 6
Geometry — Angles
In triangle ABC, angle A = 55° and angle B = 72°. What is the measure of angle C?
Problem 7
Statistics — Mean
The mean of five numbers is 12. If four of the numbers are 8, 10, 15, and 17, what is the fifth number?
Problem 8
Probability
A fair six-sided die is rolled twice. What is the probability that both rolls show an even number?
Problem 9
Counting & Combinations
A committee of 2 students is chosen from a group of 6 students. How many different committees are possible?
Problem 10
Rate — Distance/Time
Carlos drives 120 miles at 60 mph, then 80 miles at 40 mph. What is his total travel time in hours?
Problem 11
Number Theory — Primes
How many prime numbers are between 20 and 40?
Problem 12
Algebra — Linear Equations
If 3x + 7 = 28, what is the value of 2x?
Problem 13
Patterns & Sequences
The arithmetic sequence begins: 3, 7, 11, 15, … What is the 20th term?
Problem 14
Work Problems
Pipe A fills a tank in 6 hours. Pipe B fills the same tank in 3 hours. If both pipes are open together, how many hours does it take to fill the tank?
Problem 15
3D Geometry — Volume
A rectangular box has dimensions 4 cm × 5 cm × 6 cm. What is its volume in cubic centimeters?
Problem 16
Sets & Venn Diagrams
In a class of 30 students, 18 play soccer and 14 play basketball. If 6 students play both sports, how many students play neither?
Problem 17
Exponents & Powers
What is the value of 2³ × 2⁴ ÷ 2⁵?
Problem 18
Coordinate Geometry
What is the distance between points (1, 2) and (4, 6) on the coordinate plane?
Problem 19
Logic & Word Puzzles
Today is Wednesday. What day of the week will it be 100 days from today?
Problem 20
Geometry — Circle Area
A circle has a circumference of 20π cm. What is its area in square centimeters? (Use π ≈ 3.14159…)
0/20
Final Score
📚 Answer Key & Explanations
Q1
B · 4
Integers <50 divisible by both 3 and 4
Divisible by both 3 and 4 means divisible by LCM(3,4) = 12. Multiples of 12 less than 50: 12, 24, 36, 48 — that is 4 numbers. Answer: B.
Q2
E · 5/7
Largest fraction among five choices
Convert to decimals: 5/8 = 0.625, 7/11 ≈ 0.636, 3/5 = 0.600, 2/3 ≈ 0.667, 5/7 ≈ 0.714. The largest value is 5/7. Answer: E.
Q3
C · 25%
Percent increase from 800 to 1000
Percent increase = (1000 − 800)/800 × 100% = 200/800 × 100% = 25%. Answer: C.
Q4
C · 6
Sugar needed when flour:sugar = 5:2 and flour = 15
Flour/Sugar = 5/2, so Sugar = Flour × (2/5) = 15 × 2/5 = 6 cups. Answer: C.
Q5
B · 77
Area of rectangle with perimeter 36 and length 11
Perimeter = 2(l+w) → 36 = 2(11+w) → 18 = 11+w → w = 7. Area = 11 × 7 = 77 sq cm. Answer: B.
Q6
B · 53°
Third angle of triangle with 55° and 72°
Angles in a triangle sum to 180°. Angle C = 180° − 55° − 72° = 53°. Answer: B.
Q7
B · 10
Fifth number when mean of 5 numbers is 12
Total sum = 5 × 12 = 60. Known sum = 8+10+15+17 = 50. Fifth number = 60 − 50 = 10. Answer: B.
Q8
B · 1/4
P(both rolls even) on a fair six-sided die
P(even on one roll) = 3/6 = 1/2. Rolls are independent, so P(both even) = (1/2) × (1/2) = 1/4. Answer: B.
Q9
B · 15
Committees of 2 from 6 students
C(6,2) = 6!/(2!·4!) = (6×5)/(2×1) = 15. Answer: B.
Q10
C · 4
Total travel time: 120 mi at 60 mph + 80 mi at 40 mph
Time₁ = 120/60 = 2 hr. Time₂ = 80/40 = 2 hr. Total = 2 + 2 = 4 hours. Answer: C.
Q11
B · 4
Prime numbers between 20 and 40
Check each: 21=3×7✗, 22✗, 23 ✓, 24✗, 25✗, 26✗, 27✗, 28✗, 29 ✓, 30✗, 31 ✓, 32✗, 33✗, 34✗, 35✗, 36✗, 37 ✓, 38✗, 39✗. Primes: 23, 29, 31, 37 — 4 primes. Answer: B.
Q12
C · 14
Value of 2x when 3x + 7 = 28
3x + 7 = 28 → 3x = 21 → x = 7. So 2x = 2 × 7 = 14. Answer: C.
Q13
C · 79
20th term of sequence 3, 7, 11, 15, …
First term a₁ = 3, common difference d = 4. a₂₀ = 3 + (20−1)×4 = 3 + 76 = 79. Answer: C.
Q14
B · 2
Combined time for Pipe A (6 hr) and Pipe B (3 hr)
Rate A = 1/6 tank/hr, Rate B = 1/3 tank/hr. Combined rate = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 tank/hr. Time = 1 ÷ (1/2) = 2 hours. Answer: B.
Q15
C · 120
Volume of box 4 × 5 × 6 cm
Volume = l × w × h = 4 × 5 × 6 = 120 cubic cm. Answer: C.
Q16
B · 4
Students playing neither sport (soccer 18, basketball 14, both 6, total 30)
By inclusion-exclusion: students playing at least one sport = 18 + 14 − 6 = 26. Neither = 30 − 26 = 4. Answer: B.
Q17
B · 4
2³ × 2⁴ ÷ 2⁵
Using exponent rules: 2³ × 2⁴ ÷ 2⁵ = 2^(3+4−5) = 2² = 4. Answer: B.
Q18
C · 5
Distance between (1,2) and (4,6)
Distance = √[(4−1)² + (6−2)²] = √[9 + 16] = √25 = 5. Answer: C.
Q19
C · Friday
Day 100 days after Wednesday
100 ÷ 7 = 14 remainder 2. Wednesday + 2 days = Friday. Answer: C.
Q20
C · 100π
Area of circle with circumference 20π
Circumference = 2πr = 20π → r = 10. Area = πr² = π(10)² = 100π sq cm. Answer: C.