2013
2013 AMC10 Paper & Solutions Pick Paper A
25 questions with solutions, strong algebra word problems, geometry covers Pythagorean theorem.
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40 Minutes
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Exam Overview
Exam Overview
This exam emphasizes algebra, geometry, word problems. Overall difficulty is moderate.
Difficulty
- EasyQ1-10
- MediumQ11-20
- HardQ21-25
Topics
- Algebra40%
- Geometry25%
- Number Theory15%
- Combinatorics20%
Awards
AIME Qualification
AIME Qualification (Top 2.5%)
103+
Honor Roll
Honor Roll (Top 5%)
98+
Achievement Roll
Grade 6 and below · 15+ points
89+
Sample Problems
2013 AMC10 Sample Problems (12 questions)
Sample reference problems by difficulty — click an option to check your answer
Q1EasyPolynomial
If f(x) = 2x² - 6x + 2, find f(3).
Steps
1f(3) = 2×3² - 6×3 + 2
2= 2×9 - 18 + 2 = 2
Answer: DSubstitute, then exponentiate, multiply/divide, add/subtract
Q3EasyArithmetic Seq
Arithmetic sequence: a₁=4, d=5, find the 11-th term.
Steps
1aₙ = a₁ + (n-1)d
2a_11 = 4 + 10×5 = 54
Answer: EGeneral term: aₙ=a₁+(n-1)d
Q5EasyExponent
If 3ˣ = 81, find x.
Steps
181 = 3^4
2so x = 4
Answer: AConvert to same base, compare exponents
Q7EasyVieta's
Sum of roots of x² - 8x + 5 = 0?
Steps
1Vieta's:sum of roots = -(-8)/1 = 8
2product of roots = 5
Answer: BFor x²-px+q=0, sum of roots = p
Q10MediumDivisibility
How many integers from 1 to 43 are divisible by 4?
Steps
1⌊43 / 4⌋ = 10
Answer: CCount divisible by k: ⌊n/k⌋
Q12MediumDiagonals
How many diagonals does a regular 8-gon have?
Steps
1diagonals = n(n-3)/2
2= 8×5/2 = 20
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 4^5 divided by 6.
Steps
1compute 4^5 mod 6
2simplify step by step (modular arithmetic)
3=4
Answer: BModular arithmetic simplifies large exponents
Q22HardCompleting Sq
Find the minimum of f(x) = x² - 6x + 13.
Steps
1complete the square:f(x) = (x - 3)² + 4
2when x = 3 , min at 4
Answer: CCompleting the square for quadratic extrema
Q24HardInclusion-Excl
5 distinct balls into 3 distinct boxes, each ≥1 ball, how many ways?
Steps
1total 3^5 = 243
2subtract empty boxes: -C(3,1)×2^5 = -96
3add back 2 empty boxes: +C(3,2)×1 = +3
4total 243 - 96 + 3 = 150
Answer: DInclusion-exclusion handles 'at least'
Q25HardLaw of Cosines
In △ABC, a=6, b=7, cos C=1/2, find c².
Steps
1Law of cosines c² = a² + b² - 2ab·cos C
2= 36 + 49 - 42 = 43
Answer: ELaw of cosines is key for solving triangles
Q1EasyPolynomial
If f(x) = 2x² - 8x + 2, find f(3).
Steps
1f(3) = 2×3² - 8×3 + 2
2= 2×9 - 24 + 2 = -4
Answer: DSubstitute, then exponentiate, multiply/divide, add/subtract
Q3EasyArithmetic Seq
Arithmetic sequence: a₁=6, d=5, find the 11-th term.
Steps
1aₙ = a₁ + (n-1)d
2a_11 = 6 + 10×5 = 56
Answer: EGeneral term: aₙ=a₁+(n-1)d
Q5EasyExponent
If 3ˣ = 81, find x.
Steps
181 = 3^4
2so x = 4
Answer: AConvert to same base, compare exponents
Q7EasyVieta's
Sum of roots of x² - 10x + 5 = 0?
Steps
1Vieta's:sum of roots = -(-10)/1 = 10
2product of roots = 5
Answer: BFor x²-px+q=0, sum of roots = p
Q10MediumDivisibility
How many integers from 1 to 45 are divisible by 4?
Steps
1⌊45 / 4⌋ = 11
Answer: CCount divisible by k: ⌊n/k⌋
Q12MediumDiagonals
How many diagonals does a regular 10-gon have?
Steps
1diagonals = n(n-3)/2
2= 10×7/2 = 35
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 4^5 divided by 6.
Steps
1compute 4^5 mod 6
2simplify step by step (modular arithmetic)
3=4
Answer: BModular arithmetic simplifies large exponents
Q22HardCompleting Sq
Find the minimum of f(x) = x² - 6x + 15.
Steps
1complete the square:f(x) = (x - 3)² + 6
2when x = 3 , min at 6
Answer: CCompleting the square for quadratic extrema
Q24HardInclusion-Excl
5 distinct balls into 3 distinct boxes, each ≥1 ball, how many ways?
Steps
1total 3^5 = 243
2subtract empty boxes: -C(3,1)×2^5 = -96
3add back 2 empty boxes: +C(3,2)×1 = +3
4total 243 - 96 + 3 = 150
Answer: DInclusion-exclusion handles 'at least'
Q25HardLaw of Cosines
In △ABC, a=8, b=7, cos C=1/2, find c².
Steps
1Law of cosines c² = a² + b² - 2ab·cos C
2= 64 + 49 - 56 = 57
Answer: ELaw of cosines is key for solving triangles
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