2012
2012 AMC12 Paper & Solutions Pick Paper A
25 questions with solutions, harder combinatorics, geometry focuses on area transformations.
Free 2012 PDF Download
Download 2012 PDF 25 Questions
75 Minutes
Max Score 150
No Calculator
Exam Overview
Exam Overview
This exam emphasizes geometry, combinatorics, area calculations. Overall difficulty is moderate.
Difficulty
- EasyQ1-10
- MediumQ11-20
- HardQ21-25
Topics
- Algebra30%
- Geometry30%
- Number Theory20%
- Combinatorics20%
Awards
AIME Qualification
AIME Qualification (Top 2.5%)
85+
Honor Roll
Honor Roll (Top 5%)
79+
Achievement Roll
Grade 10 and below
60+
Sample Problems
2012 AMC12 Sample Problems (12 questions)
Sample reference problems by difficulty — click an option to check your answer
Q1EasyPolynomial
If f(x) = 2x² - 5x + 1, find f(2).
Steps
1f(2) = 2×2² - 5×2 + 1
2= 2×4 - 10 + 1 = -1
Answer: CSubstitute, then exponentiate, multiply/divide, add/subtract
Q3EasyArithmetic Seq
Arithmetic sequence: a₁=3, d=4, find the 10-th term.
Steps
1aₙ = a₁ + (n-1)d
2a_10 = 3 + 9×4 = 39
Answer: DGeneral term: aₙ=a₁+(n-1)d
Q5EasyExponent
If 2ˣ = 8, find x.
Steps
18 = 2^3
2so x = 3
Answer: EConvert to same base, compare exponents
Q7EasyVieta's
Sum of roots of x² - 7x + 4 = 0?
Steps
1Vieta's:sum of roots = -(-7)/1 = 7
2product of roots = 4
Answer: AFor x²-px+q=0, sum of roots = p
Q10MediumDivisibility
How many integers from 1 to 42 are divisible by 3?
Steps
1⌊42 / 3⌋ = 14
Answer: BCount 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: Cn-gon diagonals: n(n-3)/2
Q15MediumCombination
Compute C(8, 3).
Steps
1C(n,3) = n(n-1)(n-2)/6
2= 8×7×6/6 = 56
Answer: DCombination: C(n,k)=n!/(k!(n-k)!)
Q17MediumPythagorean
Right triangle legs 6 and 8, find the hypotenuse.
Steps
1c² = 6² + 8² = 36 + 64 = 100
2c = √100 = 10
Answer: EPythagorean theorem a²+b²=c²
Q20HardModular Arith
Find the remainder of 3^4 divided by 5.
Steps
1compute 3^4 mod 5
2simplify step by step (modular arithmetic)
3=1
Answer: AModular arithmetic simplifies large exponents
Q22HardCompleting Sq
Find the minimum of f(x) = x² - 4x + 7.
Steps
1complete the square:f(x) = (x - 2)² + 3
2when x = 2 , min at 3
Answer: BCompleting 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: CInclusion-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: DLaw of cosines is key for solving triangles
Q1EasyPolynomial
If f(x) = 2x² - 7x + 1, find f(2).
Steps
1f(2) = 2×2² - 7×2 + 1
2= 2×4 - 14 + 1 = -5
Answer: CSubstitute, then exponentiate, multiply/divide, add/subtract
Q3EasyArithmetic Seq
Arithmetic sequence: a₁=5, d=4, find the 10-th term.
Steps
1aₙ = a₁ + (n-1)d
2a_10 = 5 + 9×4 = 41
Answer: DGeneral term: aₙ=a₁+(n-1)d
Q5EasyExponent
If 2ˣ = 8, find x.
Steps
18 = 2^3
2so x = 3
Answer: EConvert to same base, compare exponents
Q7EasyVieta's
Sum of roots of x² - 9x + 4 = 0?
Steps
1Vieta's:sum of roots = -(-9)/1 = 9
2product of roots = 4
Answer: AFor x²-px+q=0, sum of roots = p
Q10MediumDivisibility
How many integers from 1 to 44 are divisible by 3?
Steps
1⌊44 / 3⌋ = 14
Answer: BCount 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: Cn-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: DCombination: 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: EPythagorean theorem a²+b²=c²
Q20HardModular Arith
Find the remainder of 3^4 divided by 5.
Steps
1compute 3^4 mod 5
2simplify step by step (modular arithmetic)
3=1
Answer: AModular arithmetic simplifies large exponents
Q22HardCompleting Sq
Find the minimum of f(x) = x² - 4x + 9.
Steps
1complete the square:f(x) = (x - 2)² + 5
2when x = 2 , min at 5
Answer: BCompleting 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: CInclusion-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: DLaw of cosines is key for solving triangles
All Years
All Years Navigation
All AMC12 exam years from 2000-2025, click to view
202525 Q & Sol.
202425 Q & Sol.
202325 Q & Sol.
202225 Q & Sol.
202125 Q & Sol.
202025 Q & Sol.
201925 Q & Sol.
201825 Q & Sol.
201725 Q & Sol.
201625 Q & Sol.
201525 Q & Sol.
201425 Q & Sol.
201325 Q & Sol.
201225 Q & Sol.
201125 Q & Sol.
201025 Q & Sol.
200925 Q & Sol.
200825 Q & Sol.
200725 Q & Sol.
200625 Q & Sol.
200525 Q & Sol.
200425 Q & Sol.
200325 Q & Sol.
200225 Q & Sol.
200125 Q & Sol.
200025 Q & Sol.