2023
2023 AMC10 Paper & Solutions Pick Paper A
25 questions with step-by-step solutions, topic summaries and method comparisons. Hard combinatorics, similar triangles in geometry.
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25 Questions
40 Minutes
Max Score 25
No Calculator
Exam Overview
Exam Overview
This exam emphasizes geometry, combinatorics. Overall difficulty is moderate.
Difficulty
- EasyQ1-10
- MediumQ11-20
- HardQ21-25
Topics
- Algebra35%
- Geometry30%
- Number Theory15%
- Combinatorics20%
Awards
AIME Qualification
AIME Qualification (Top 2.5%)
113+
Honor Roll
Honor Roll (Top 5%)
100+
Achievement Roll
Grade 6 and below · 15+ points
90+
Sample Problems
2023 AMC10 Sample Problems (12 questions)
Sample reference problems by difficulty — click an option to check your answer
Q1EasyPolynomial
If f(x) = 2x² - 6x + 3, find f(5).
Steps
1f(5) = 2×5² - 6×5 + 3
2= 2×25 - 30 + 3 = 23
Answer: DSubstitute, then exponentiate, multiply/divide, add/subtract
Q3EasyArithmetic Seq
Arithmetic sequence: a₁=6, d=5, find the 15-th term.
Steps
1aₙ = a₁ + (n-1)d
2a_15 = 6 + 14×5 = 76
Answer: EGeneral term: aₙ=a₁+(n-1)d
Q5EasyExponent
If 4ˣ = 4096, find x.
Steps
14096 = 4^6
2so x = 6
Answer: AConvert to same base, compare exponents
Q7EasyVieta's
Sum of roots of x² - 8x + 7 = 0?
Steps
1Vieta's:sum of roots = -(-8)/1 = 8
2product of roots = 7
Answer: BFor x²-px+q=0, sum of roots = p
Q10MediumDivisibility
How many integers from 1 to 33 are divisible by 5?
Steps
1⌊33 / 5⌋ = 6
Answer: CCount 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: 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 6^6 divided by 8.
Steps
1compute 6^6 mod 8
2simplify step by step (modular arithmetic)
3=0
Answer: BModular arithmetic simplifies large exponents
Q22HardCompleting Sq
Find the minimum of f(x) = x² - 10x + 30.
Steps
1complete the square:f(x) = (x - 5)² + 5
2when x = 5 , min at 5
Answer: CCompleting 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: DInclusion-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: ELaw of cosines is key for solving triangles
Q1EasyPolynomial
If f(x) = 2x² - 8x + 3, find f(5).
Steps
1f(5) = 2×5² - 8×5 + 3
2= 2×25 - 40 + 3 = 13
Answer: DSubstitute, then exponentiate, multiply/divide, add/subtract
Q3EasyArithmetic Seq
Arithmetic sequence: a₁=8, d=5, find the 15-th term.
Steps
1aₙ = a₁ + (n-1)d
2a_15 = 8 + 14×5 = 78
Answer: EGeneral term: aₙ=a₁+(n-1)d
Q5EasyExponent
If 4ˣ = 4096, find x.
Steps
14096 = 4^6
2so x = 6
Answer: AConvert to same base, compare exponents
Q7EasyVieta's
Sum of roots of x² - 10x + 7 = 0?
Steps
1Vieta's:sum of roots = -(-10)/1 = 10
2product of roots = 7
Answer: BFor x²-px+q=0, sum of roots = p
Q10MediumDivisibility
How many integers from 1 to 35 are divisible by 5?
Steps
1⌊35 / 5⌋ = 7
Answer: CCount divisible by k: ⌊n/k⌋
Q12MediumDiagonals
How many diagonals does a regular 14-gon have?
Steps
1diagonals = n(n-3)/2
2= 14×11/2 = 77
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 6^6 divided by 8.
Steps
1compute 6^6 mod 8
2simplify step by step (modular arithmetic)
3=0
Answer: BModular arithmetic simplifies large exponents
Q22HardCompleting Sq
Find the minimum of f(x) = x² - 10x + 32.
Steps
1complete the square:f(x) = (x - 5)² + 7
2when x = 5 , min at 7
Answer: CCompleting 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: DInclusion-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: ELaw of cosines is key for solving triangles
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