2012
2012 AMC10 Paper & Solutions Pick Paper A
25 questions with solutions, harder combinatorics, geometry focuses on area transformations.
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Download 2012 PDF 25 Questions
40 Minutes
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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%)
115+
Honor Roll
Honor Roll (Top 5%)
97+
Achievement Roll
Grade 6 and below · 15+ points
88+
Sample Problems
2012 AMC10 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
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