2014
2014 AMC8 Paper & Solutions Pick
25 questions with solutions, number theory covers divisor count theorem, basic probability.
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Exam Overview
Exam Overview
# of divisors, Probability.
Difficulty
- EasyQ1-10
- MediumQ11-20
- HardQ21-25
Topics
- Algebra35%
- Geometry25%
- Number Theory20%
- Combinatorics20%
Awards
Distinguished Honor Roll
Distinguished Honor Roll (Top 1%)
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Honor Roll
Honor Roll (Top 5%)
16+
Achievement Roll
Grade 6 and below · 15+ points
15+
Sample Problems
2014 AMC8 Sample Problems (12 questions)
Sample reference problems by difficulty — click an option to check your answer
Q2EasyFractions
Calculate the value of 3/5 × 10/9.
Steps
3/5×10/9=30/45=2/3
Answer: BCross-cancel between the numerators and denominators.
Q4EasyApply the relevant mathematical concept
What is the largest integer x such that 3x-1 < 14?
Steps
13x<15 → x<5
Largest integer = 4
Answer: BTake the largest integer in the inequality's solution set.
Q6EasyKey concept: factors
How many positive divisors does 48 have?
Steps
48=2⁴×3
Number of divisors=(4+1)(1+1)=10
Answer: BUse the formula for the number of divisors.
Q9EasyArea
A rectangle has area 72, and its length is twice its width. What is the width?
Steps
2w×w=72 → w²=36 → w=6
Answer: EArea = length × width.
Q11MediumProbability
A number is chosen at random from 1 to 20. What is the probability that it is even or a multiple of 5?
Steps
There are 10 even numbers; 4 multiples of 5 (5, 10, 15, 20)
Overlap (both even and a multiple of 5): 10 and 20, so 2 numbers
10 + 4 − 2 = 12, probability = 12/20 = 3/5
Answer: EFor "or," use inclusion-exclusion: A + B − overlap.
Q13MediumKey concept: equations
A train is 200 meters long and takes 35 seconds to cross a bridge that is 500 meters long. What is the train's speed in m/s?
Steps
Total distance=200+500=700 meters
Speed=700/35=20 m/s
Answer: CDistance for a train crossing a bridge = train length + bridge length.
Q16MediumGeometry
An isosceles triangle has legs of length 10 and a base of 12. What is its area?
Steps
Half of the base=6, height=√(100-36)=8
Area=1/2×12×8=48
Answer: CIn an isosceles triangle, the altitude, median, and angle bisector to the base coincide.
Q18MediumNumber Theory
Find the value of GCD(168, 198).
Steps
Euclidean algorithm: 198=1×168+30
168=5×30+18
30=1×18+12
18=1×12+6
12=2×6+0
GCD=6
Answer: EThe Euclidean algorithm is the standard method for finding the GCD.
Q21HardKey concept: factors
How many positive divisors of 720 are even?
Steps
720=2⁴×3²×5
Total divisors=(4+1)(2+1)(1+1)=30
Odd divisors (without any factor of 2)=1×3×2=6
Even divisors=30-6=24
Answer: DEven divisors = total divisors − odd divisors.
Q22HardKey concept: permutations
Six different books are arranged on a shelf, where the math book and the Chinese book must be next to each other. How many arrangements are there?
Steps
Bundling method: 5!×2=240
Answer: CUse the bundling method for adjacency problems.
Q24HardArea
The inscribed circle of a square has area 154 (π≈22/7). What is the area of the square?
Steps
πr²=154 → r=7
Side length of the square=2r=14
Area=196
Answer: DThe diameter of the inscribed circle equals the side length of the square.
Q25HardCombinatorics
Choosing 2 numbers from 1 to 15 so that their sum is even, how many ways are there?
Steps
8 odd numbers and 7 even numbers
Both odd C(8,2)=28, both even C(7,2)=21
28+21=49
Answer: CThe sum of two numbers is even when both are odd or both are even.
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