2018
2018 AMC12 Paper & Solutions Pick Paper A
25 complete solutions, geometry covers Pythagorean theorem and area transforms, algebra focuses on inequalities and functions.
Scan to get free 2018 PDF + solutions
25 Questions
75 Minutes
Max Score 150
No Calculator
Exam Overview
Exam Overview
This exam emphasizes geometry, area calculations. Overall difficulty is moderate.
Difficulty
- EasyQ1-10
- MediumQ11-20
- HardQ21-25
Topics
- Algebra35%
- Geometry35%
- Number Theory15%
- Combinatorics15%
Awards
AIME Qualification
AIME Qualification (Top 2.5%)
91+
Honor Roll
Honor Roll (Top 5%)
77+
Achievement Roll
Grade 10 and below
60+
Sample Problems
2018 AMC12 Sample Problems (12 questions)
Sample reference problems by difficulty — click an option to check your answer
Q1EasyPolynomial
If f(x) = 2x² - 6x + 1, find f(4).
Steps
1f(4) = 2×4² - 6×4 + 1
2= 2×16 - 24 + 1 = 9
Answer: DSubstitute, then exponentiate, multiply/divide, add/subtract
Q3EasyArithmetic Seq
Arithmetic sequence: a₁=5, d=5, find the 10-th term.
Steps
1aₙ = a₁ + (n-1)d
2a_10 = 5 + 9×5 = 50
Answer: EGeneral term: aₙ=a₁+(n-1)d
Q5EasyExponent
If 2ˣ = 32, find x.
Steps
132 = 2^5
2so x = 5
Answer: AConvert to same base, compare exponents
Q7EasyVieta's
Sum of roots of x² - 8x + 6 = 0?
Steps
1Vieta's:sum of roots = -(-8)/1 = 8
2product of roots = 6
Answer: BFor x²-px+q=0, sum of roots = p
Q10MediumDivisibility
How many integers from 1 to 48 are divisible by 3?
Steps
1⌊48 / 3⌋ = 16
Answer: CCount divisible by k: ⌊n/k⌋
Q12MediumDiagonals
How many diagonals does a regular 7-gon have?
Steps
1diagonals = n(n-3)/2
2= 7×4/2 = 14
Answer: Dn-gon diagonals: n(n-3)/2
Q15MediumCombination
Compute C(9, 3).
Steps
1C(n,3) = n(n-1)(n-2)/6
2= 9×8×7/6 = 84
Answer: ECombination: C(n,k)=n!/(k!(n-k)!)
Q17MediumPythagorean
Right triangle legs 8 and 15, find the hypotenuse.
Steps
1c² = 8² + 15² = 64 + 225 = 289
2c = √289 = 17
Answer: APythagorean theorem a²+b²=c²
Q20HardModular Arith
Find the remainder of 5^4 divided by 7.
Steps
1compute 5^4 mod 7
2simplify step by step (modular arithmetic)
3=2
Answer: BModular arithmetic simplifies large exponents
Q22HardCompleting Sq
Find the minimum of f(x) = x² - 8x + 19.
Steps
1complete the square:f(x) = (x - 4)² + 3
2when x = 4 , min at 3
Answer: CCompleting the square for quadratic extrema
Q24HardInclusion-Excl
4 distinct balls into 3 distinct boxes, each ≥1 ball, how many ways?
Steps
1total 3^4 = 81
2subtract empty boxes: -C(3,1)×2^4 = -48
3add back 2 empty boxes: +C(3,2)×1 = +3
4total 81 - 48 + 3 = 36
Answer: DInclusion-exclusion handles 'at least'
Q25HardLaw of Cosines
In △ABC, a=5, b=6, cos C=1/2, find c².
Steps
1Law of cosines c² = a² + b² - 2ab·cos C
2= 25 + 36 - 30 = 31
Answer: ELaw of cosines is key for solving triangles
Q1EasyPolynomial
If f(x) = 2x² - 8x + 1, find f(4).
Steps
1f(4) = 2×4² - 8×4 + 1
2= 2×16 - 32 + 1 = 1
Answer: DSubstitute, then exponentiate, multiply/divide, add/subtract
Q3EasyArithmetic Seq
Arithmetic sequence: a₁=7, d=5, find the 10-th term.
Steps
1aₙ = a₁ + (n-1)d
2a_10 = 7 + 9×5 = 52
Answer: EGeneral term: aₙ=a₁+(n-1)d
Q5EasyExponent
If 2ˣ = 32, find x.
Steps
132 = 2^5
2so x = 5
Answer: AConvert to same base, compare exponents
Q7EasyVieta's
Sum of roots of x² - 10x + 6 = 0?
Steps
1Vieta's:sum of roots = -(-10)/1 = 10
2product of roots = 6
Answer: BFor x²-px+q=0, sum of roots = p
Q10MediumDivisibility
How many integers from 1 to 50 are divisible by 3?
Steps
1⌊50 / 3⌋ = 16
Answer: CCount divisible by k: ⌊n/k⌋
Q12MediumDiagonals
How many diagonals does a regular 9-gon have?
Steps
1diagonals = n(n-3)/2
2= 9×6/2 = 27
Answer: Dn-gon diagonals: n(n-3)/2
Q15MediumCombination
Compute C(11, 3).
Steps
1C(n,3) = n(n-1)(n-2)/6
2= 11×10×9/6 = 165
Answer: ECombination: C(n,k)=n!/(k!(n-k)!)
Q17MediumPythagorean
Right triangle legs 3 and 4, find the hypotenuse.
Steps
1c² = 3² + 4² = 9 + 16 = 25
2c = √25 = 5
Answer: APythagorean theorem a²+b²=c²
Q20HardModular Arith
Find the remainder of 5^4 divided by 7.
Steps
1compute 5^4 mod 7
2simplify step by step (modular arithmetic)
3=2
Answer: BModular arithmetic simplifies large exponents
Q22HardCompleting Sq
Find the minimum of f(x) = x² - 8x + 21.
Steps
1complete the square:f(x) = (x - 4)² + 5
2when x = 4 , min at 5
Answer: CCompleting the square for quadratic extrema
Q24HardInclusion-Excl
4 distinct balls into 3 distinct boxes, each ≥1 ball, how many ways?
Steps
1total 3^4 = 81
2subtract empty boxes: -C(3,1)×2^4 = -48
3add back 2 empty boxes: +C(3,2)×1 = +3
4total 81 - 48 + 3 = 36
Answer: DInclusion-exclusion handles 'at least'
Q25HardLaw of Cosines
In △ABC, a=7, b=6, cos C=1/2, find c².
Steps
1Law of cosines c² = a² + b² - 2ab·cos C
2= 49 + 36 - 42 = 43
Answer: ELaw of cosines is key for solving triangles
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