2021
2021 AMC10 Paper & Solutions Pick Paper A
Online exam year, 25 complete solutions. Many practical application problems (shopping discounts, trip planning).
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25 Questions
40 Minutes
Max Score 25
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Exam Overview
Exam Overview
This exam emphasizes multiple math topics. 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%)
111+
Honor Roll
Honor Roll (Top 5%)
98+
Achievement Roll
Grade 6 and below · 15+ points
88+
Sample Problems
2021 AMC10 Sample Problems (12 questions)
Sample reference problems by difficulty — click an option to check your answer
Q1EasyPolynomial
If f(x) = 2x² - 4x + 1, find f(3).
Steps
1f(3) = 2×3² - 4×3 + 1
2= 2×9 - 12 + 1 = 7
Answer: BSubstitute, then exponentiate, multiply/divide, add/subtract
Q3EasyArithmetic Seq
Arithmetic sequence: a₁=4, d=3, find the 13-th term.
Steps
1aₙ = a₁ + (n-1)d
2a_13 = 4 + 12×3 = 40
Answer: CGeneral term: aₙ=a₁+(n-1)d
Q5EasyExponent
If 2ˣ = 16, find x.
Steps
116 = 2^4
2so x = 4
Answer: DConvert to same base, compare exponents
Q7EasyVieta's
Sum of roots of x² - 6x + 5 = 0?
Steps
1Vieta's:sum of roots = -(-6)/1 = 6
2product of roots = 5
Answer: EFor x²-px+q=0, sum of roots = p
Q10MediumDivisibility
How many integers from 1 to 31 are divisible by 3?
Steps
1⌊31 / 3⌋ = 10
Answer: ACount 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: Bn-gon diagonals: n(n-3)/2
Q15MediumCombination
Compute C(7, 3).
Steps
1C(n,3) = n(n-1)(n-2)/6
2= 7×6×5/6 = 35
Answer: CCombination: 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: DPythagorean theorem a²+b²=c²
Q20HardModular Arith
Find the remainder of 4^4 divided by 6.
Steps
1compute 4^4 mod 6
2simplify step by step (modular arithmetic)
3=4
Answer: EModular arithmetic simplifies large exponents
Q22HardCompleting Sq
Find the minimum of f(x) = x² - 6x + 12.
Steps
1complete the square:f(x) = (x - 3)² + 3
2when x = 3 , min at 3
Answer: ACompleting 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: BInclusion-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: CLaw of cosines is key for solving triangles
Q1EasyPolynomial
If f(x) = 2x² - 6x + 1, find f(3).
Steps
1f(3) = 2×3² - 6×3 + 1
2= 2×9 - 18 + 1 = 1
Answer: BSubstitute, then exponentiate, multiply/divide, add/subtract
Q3EasyArithmetic Seq
Arithmetic sequence: a₁=6, d=3, find the 13-th term.
Steps
1aₙ = a₁ + (n-1)d
2a_13 = 6 + 12×3 = 42
Answer: CGeneral term: aₙ=a₁+(n-1)d
Q5EasyExponent
If 2ˣ = 16, find x.
Steps
116 = 2^4
2so x = 4
Answer: DConvert 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: EFor x²-px+q=0, sum of roots = p
Q10MediumDivisibility
How many integers from 1 to 33 are divisible by 3?
Steps
1⌊33 / 3⌋ = 11
Answer: ACount divisible by k: ⌊n/k⌋
Q12MediumDiagonals
How many diagonals does a regular 12-gon have?
Steps
1diagonals = n(n-3)/2
2= 12×9/2 = 54
Answer: Bn-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: CCombination: 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: DPythagorean theorem a²+b²=c²
Q20HardModular Arith
Find the remainder of 4^4 divided by 6.
Steps
1compute 4^4 mod 6
2simplify step by step (modular arithmetic)
3=4
Answer: EModular arithmetic simplifies large exponents
Q22HardCompleting Sq
Find the minimum of f(x) = x² - 6x + 14.
Steps
1complete the square:f(x) = (x - 3)² + 5
2when x = 3 , min at 5
Answer: ACompleting 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: BInclusion-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: CLaw of cosines is key for solving triangles
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