2019
2019 AMC10 Paper & Solutions Pick Paper A
25 complete solutions, algebra focuses on equation applications, geometry covers circle properties. Combinatorics introduces recursion.
Scan to get free 2019 PDF + solutions
25 Questions
40 Minutes
Max Score 25
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Exam Overview
Exam Overview
This exam emphasizes algebra, combinatorics, equations. 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%)
109+
Honor Roll
Honor Roll (Top 5%)
96+
Achievement Roll
Grade 6 and below · 15+ points
89+
Sample Problems
2019 AMC10 Sample Problems (12 questions)
Sample reference problems by difficulty — click an option to check your answer
Q1EasyPolynomial
If f(x) = 2x² - 7x + 2, find f(5).
Steps
1f(5) = 2×5² - 7×5 + 2
2= 2×25 - 35 + 2 = 17
Answer: ESubstitute, then exponentiate, multiply/divide, add/subtract
Q3EasyArithmetic Seq
Arithmetic sequence: a₁=6, d=6, find the 11-th term.
Steps
1aₙ = a₁ + (n-1)d
2a_11 = 6 + 10×6 = 66
Answer: AGeneral term: aₙ=a₁+(n-1)d
Q5EasyExponent
If 3ˣ = 729, find x.
Steps
1729 = 3^6
2so x = 6
Answer: BConvert to same base, compare exponents
Q7EasyVieta's
Sum of roots of x² - 9x + 7 = 0?
Steps
1Vieta's:sum of roots = -(-9)/1 = 9
2product of roots = 7
Answer: CFor x²-px+q=0, sum of roots = p
Q10MediumDivisibility
How many integers from 1 to 49 are divisible by 4?
Steps
1⌊49 / 4⌋ = 12
Answer: DCount 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: En-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: ACombination: 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: BPythagorean theorem a²+b²=c²
Q20HardModular Arith
Find the remainder of 6^5 divided by 8.
Steps
1compute 6^5 mod 8
2simplify step by step (modular arithmetic)
3=0
Answer: CModular arithmetic simplifies large exponents
Q22HardCompleting Sq
Find the minimum of f(x) = x² - 10x + 29.
Steps
1complete the square:f(x) = (x - 5)² + 4
2when x = 5 , min at 4
Answer: DCompleting 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: EInclusion-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: ALaw of cosines is key for solving triangles
Q1EasyPolynomial
If f(x) = 2x² - 9x + 2, find f(5).
Steps
1f(5) = 2×5² - 9×5 + 2
2= 2×25 - 45 + 2 = 7
Answer: ESubstitute, then exponentiate, multiply/divide, add/subtract
Q3EasyArithmetic Seq
Arithmetic sequence: a₁=8, d=6, find the 11-th term.
Steps
1aₙ = a₁ + (n-1)d
2a_11 = 8 + 10×6 = 68
Answer: AGeneral term: aₙ=a₁+(n-1)d
Q5EasyExponent
If 3ˣ = 729, find x.
Steps
1729 = 3^6
2so x = 6
Answer: BConvert to same base, compare exponents
Q7EasyVieta's
Sum of roots of x² - 11x + 7 = 0?
Steps
1Vieta's:sum of roots = -(-11)/1 = 11
2product of roots = 7
Answer: CFor x²-px+q=0, sum of roots = p
Q10MediumDivisibility
How many integers from 1 to 51 are divisible by 4?
Steps
1⌊51 / 4⌋ = 12
Answer: DCount 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: En-gon diagonals: n(n-3)/2
Q15MediumCombination
Compute C(12, 3).
Steps
1C(n,3) = n(n-1)(n-2)/6
2= 12×11×10/6 = 220
Answer: ACombination: 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: BPythagorean theorem a²+b²=c²
Q20HardModular Arith
Find the remainder of 6^5 divided by 8.
Steps
1compute 6^5 mod 8
2simplify step by step (modular arithmetic)
3=0
Answer: CModular arithmetic simplifies large exponents
Q22HardCompleting Sq
Find the minimum of f(x) = x² - 10x + 31.
Steps
1complete the square:f(x) = (x - 5)² + 6
2when x = 5 , min at 6
Answer: DCompleting 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: EInclusion-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: ALaw of cosines is key for solving triangles
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