2014
2014 AMC12 Paper & Solutions Pick Paper A
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
AIME Qualification
AIME Qualification (Top 2.5%)
87+
Honor Roll
Honor Roll (Top 5%)
81+
Achievement Roll
Grade 10 and below
62+
Sample Problems
2014 AMC12 Sample Problems (12 questions)
Sample reference problems by difficulty — click an option to check your answer
Q1EasyPolynomial
If f(x) = 2x² - 7x + 3, find f(4).
Steps
1f(4) = 2×4² - 7×4 + 3
2= 2×16 - 28 + 3 = 7
Answer: ESubstitute, then exponentiate, multiply/divide, add/subtract
Q3EasyArithmetic Seq
Arithmetic sequence: a₁=5, d=6, find the 12-th term.
Steps
1aₙ = a₁ + (n-1)d
2a_12 = 5 + 11×6 = 71
Answer: AGeneral term: aₙ=a₁+(n-1)d
Q5EasyExponent
If 4ˣ = 1024, find x.
Steps
11024 = 4^5
2so x = 5
Answer: BConvert to same base, compare exponents
Q7EasyVieta's
Sum of roots of x² - 9x + 6 = 0?
Steps
1Vieta's:sum of roots = -(-9)/1 = 9
2product of roots = 6
Answer: CFor x²-px+q=0, sum of roots = p
Q10MediumDivisibility
How many integers from 1 to 44 are divisible by 5?
Steps
1⌊44 / 5⌋ = 8
Answer: DCount 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: En-gon diagonals: n(n-3)/2
Q15MediumCombination
Compute C(10, 3).
Steps
1C(n,3) = n(n-1)(n-2)/6
2= 10×9×8/6 = 120
Answer: ACombination: C(n,k)=n!/(k!(n-k)!)
Q17MediumPythagorean
Right triangle legs 7 and 24, find the hypotenuse.
Steps
1c² = 7² + 24² = 49 + 576 = 625
2c = √625 = 25
Answer: BPythagorean theorem a²+b²=c²
Q20HardModular Arith
Find the remainder of 5^6 divided by 7.
Steps
1compute 5^6 mod 7
2simplify step by step (modular arithmetic)
3=1
Answer: CModular 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: DCompleting the square for quadratic extrema
Q24HardInclusion-Excl
6 distinct balls into 3 distinct boxes, each ≥1 ball, how many ways?
Steps
1total 3^6 = 729
2subtract empty boxes: -C(3,1)×2^6 = -192
3add back 2 empty boxes: +C(3,2)×1 = +3
4total 729 - 192 + 3 = 540
Answer: EInclusion-exclusion handles 'at least'
Q25HardLaw of Cosines
In △ABC, a=7, b=8, cos C=1/2, find c².
Steps
1Law of cosines c² = a² + b² - 2ab·cos C
2= 49 + 64 - 56 = 57
Answer: ALaw of cosines is key for solving triangles
Q1EasyPolynomial
If f(x) = 2x² - 9x + 3, find f(4).
Steps
1f(4) = 2×4² - 9×4 + 3
2= 2×16 - 36 + 3 = -1
Answer: ESubstitute, then exponentiate, multiply/divide, add/subtract
Q3EasyArithmetic Seq
Arithmetic sequence: a₁=7, d=6, find the 12-th term.
Steps
1aₙ = a₁ + (n-1)d
2a_12 = 7 + 11×6 = 73
Answer: AGeneral term: aₙ=a₁+(n-1)d
Q5EasyExponent
If 4ˣ = 1024, find x.
Steps
11024 = 4^5
2so x = 5
Answer: BConvert to same base, compare exponents
Q7EasyVieta's
Sum of roots of x² - 11x + 6 = 0?
Steps
1Vieta's:sum of roots = -(-11)/1 = 11
2product of roots = 6
Answer: CFor x²-px+q=0, sum of roots = p
Q10MediumDivisibility
How many integers from 1 to 46 are divisible by 5?
Steps
1⌊46 / 5⌋ = 9
Answer: DCount divisible by k: ⌊n/k⌋
Q12MediumDiagonals
How many diagonals does a regular 11-gon have?
Steps
1diagonals = n(n-3)/2
2= 11×8/2 = 44
Answer: En-gon diagonals: n(n-3)/2
Q15MediumCombination
Compute C(12, 3).
Steps
1C(n,3) = n(n-1)(n-2)/6
2= 12×11×10/6 = 220
Answer: ACombination: C(n,k)=n!/(k!(n-k)!)
Q17MediumPythagorean
Right triangle legs 5 and 12, find the hypotenuse.
Steps
1c² = 5² + 12² = 25 + 144 = 169
2c = √169 = 13
Answer: BPythagorean theorem a²+b²=c²
Q20HardModular Arith
Find the remainder of 5^6 divided by 7.
Steps
1compute 5^6 mod 7
2simplify step by step (modular arithmetic)
3=1
Answer: CModular arithmetic simplifies large exponents
Q22HardCompleting Sq
Find the minimum of f(x) = x² - 8x + 23.
Steps
1complete the square:f(x) = (x - 4)² + 7
2when x = 4 , min at 7
Answer: DCompleting the square for quadratic extrema
Q24HardInclusion-Excl
6 distinct balls into 3 distinct boxes, each ≥1 ball, how many ways?
Steps
1total 3^6 = 729
2subtract empty boxes: -C(3,1)×2^6 = -192
3add back 2 empty boxes: +C(3,2)×1 = +3
4total 729 - 192 + 3 = 540
Answer: EInclusion-exclusion handles 'at least'
Q25HardLaw of Cosines
In △ABC, a=9, b=8, cos C=1/2, find c².
Steps
1Law of cosines c² = a² + b² - 2ab·cos C
2= 81 + 64 - 72 = 73
Answer: ALaw of cosines is key for solving triangles
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