The 2023 AMC 10 Papers A and B leaned toward algebraic equations and combinatorial counting, with geometry skewed toward computation; factoring-and-Vieta, permutation-combination, divisibility and units-digit problems appeared often, raising the demand on back-half categorization and discussion. Scan for the full papers and answer keys.
2023 AMC 10 A/B paper overview and exam highlights.
The 2023 AMC 10 was held in November 2023 in two papers (A and B), leaning on algebraic equations and combinatorial counting, with geometry tilted toward computation.
Format: 25 multiple-choice questions, 75 minutes, 150 points max; a blank answer earns 1.5 points and a wrong answer earns 0. The AIME cut-off has recently been around 100–105 (Paper A).
The 2023 paper featured factoring and Vieta, permutations and combinations, divisibility and units digits, raising the demand for casework on mid-to-late questions.
2023 AMC 10 main topic areas, weights and emphasis.
Evaluation, Vieta, word and speed problems; about 30%.
Rectangle perimeter, circle area and circumference, triangles and cylinders; about 22%.
Divisibility, remainders, GCD and units digits of powers; about 18%.
Permutations, combinations and ordered counting; about 20%.
Classical probability and dice sums; about 10%.
2023 AMC 10-style mock questions (Basic · Medium · Hard) — submit to grade and view step-by-step solutions.
If x = 5, what is the value of 2x2 - 3x + 4?
Correct answer: D
Substitute x = 5: 2(5)2 - 3(5) + 4 = 50 - 15 + 4 = 39. Answer D. Key: direct substitution.The sum of three consecutive integers is 60. What is the largest?
Correct answer: D
Let the middle integer be n: 3n = 60, so n = 20 and the largest is 21. Answer D. Key: model consecutive integers by their middle.How many positive divisors does 16 have?
Correct answer: B
16 = 24, so the number of divisors is 4+1 = 5 (namely 1,2,4,8,16). Answer B. Key: divisor count of a prime power.A rectangle has length 9 and width 3. What is its perimeter?
Correct answer: B
Perimeter = 2(9 + 3) = 2×12 = 24. Answer B. (27 is the area — do not confuse.) Key: rectangle perimeter formula.An item is marked up 15% above cost to a price of $69. What was the cost?
Correct answer: C
cost × 1.15 = 69, so cost = 69 / 1.15 = 60. Answer C. Key: price = cost × (1 + markup).A fair six-sided die is rolled once. What is the probability of rolling a multiple of 3 (i.e. 3 or 6)?
Correct answer: B
Multiples of 3 are 3 and 6 (2 of 6), so the probability is 2/6 = 1/3. Answer B. Key: classical probability = favorable / total.In how many ways can 5 different books be arranged in a row?
Correct answer: C
Permutations of 5 distinct items: 5! = 5×4×3×2×1 = 120. Answer C. Key: n distinct items have n! arrangements.What is the remainder when 2023 is divided by 9?
Correct answer: E
The digit sum is 2+0+2+3 = 7, so the remainder of 2023 divided by 9 equals 7. Answer E. Key: divisibility-by-9 test and digit sum.If 4x + 1 = 29, what is x?
Correct answer: D
4x = 29 - 1 = 28, so x = 7. Answer D. Key: isolate the variable.A circle has radius 14. What is its circumference? (Use π ≈ 22/7)
Correct answer: B
Circumference = 2πr = 2×(22/7)×14 = 44×2 = 88. Answer B. Key: circle circumference = 2πr.If x2 - 8x + 15 = 0, what is the sum of all possible values of x?
Correct answer: D
Factor: (x-3)(x-5) = 0, so x = 3 or 5, and the sum is 8. Answer D. Key: Vieta — sum of roots = -b/a = 8.What is the greatest common divisor of 48 and 72?
Correct answer: C
48 = 24×3 and 72 = 23×32, so GCD = 23×3 = 24. Answer C. Key: take the smallest power of each shared prime.A committee of 2 is chosen from 7 people. How many different committees are possible?
Correct answer: B
C(7,2) = (7×6)/(2×1) = 21. Answer B. Key: combinations are order-independent, C(n,k) = n!/(k!(n-k)!).In a triangle, two angles are 50° and 60°. What is the third angle?
Correct answer: C
Triangle angles sum to 180°, so the third angle = 180 - 50 - 60 = 70°. Answer C. Key: triangle angle-sum theorem.The average of 4 numbers is 15. After removing one number, the average of the remaining 3 is 13. What was the removed number?
Correct answer: D
Sum of 4 = 4×15 = 60; sum of 3 = 3×13 = 39; the removed number = 60 - 39 = 21. Answer D. Key: average = sum / count.What is the units digit of 22023?
Correct answer: D
The units digit of powers of 2 cycles as 2, 4, 8, 6 (period 4). Since 2023 = 4×505 + 3, the remainder is 3, corresponding to the 3rd value, so the units digit is 8. Answer D. Key: cyclic units digits of powers.One worker can finish a job in 6 hours and another in 2 hours. How long does it take them together?
Correct answer: B
Rates: 1/6 + 1/2 = 1/6 + 3/6 = 4/6 = 2/3 per hour, so together they need 1 / (2/3) = 1.5 hours. Answer B. Key: combined rate = sum of individual rates.A square and an equilateral triangle have the same perimeter. If the triangle side is 16, what is the square side?
Correct answer: C
Triangle perimeter = 3×16 = 48, so the square side = 48 / 4 = 12. Answer C. Key: equal perimeters split by side count.Two fair six-sided dice are rolled. What is the probability the sum is 9?
Correct answer: B
Sums to 9: (3,6),(4,5),(5,4),(6,3) — 4 outcomes out of 36, so 4/36 = 1/9. Answer B. Key: enumerate ordered dice pairs.A car travels 90 km at 45 km/h and returns at 90 km/h. What is the average speed for the whole trip (km/h)?
Correct answer: B
Outbound time = 90/45 = 2 h; return = 90/90 = 1 h. Total distance 180 km over 3 h gives 180/3 = 60 km/h. Answer B. Key: average speed = total distance / total time, not the average of speeds.How many positive integers less than 100 are divisible by both 3 and 5?
Correct answer: C
Divisible by both 3 and 5 means divisible by LCM(3,5) = 15. The multiples of 15 below 100 are 15, 30, 45, 60, 75, 90 — 6 of them. Answer C. Key: divisible by both ⟺ divisible by their LCM.Using the digits 1, 2, 3, 4, 5 (each at most once), how many three-digit numbers can be formed?
Correct answer: C
Hundreds digit 5 choices, tens 4, units 3, giving 5×4×3 = 60. Answer C. Key: permutations P(5,3) = 5!/(5-3)!.A right cylinder has radius 5 and height 2. What is its total surface area? (in terms of π)
Correct answer: C
Surface area = 2πr2 + 2πrh = 2π(52) + 2π(5)(2) = 50π + 20π = 70π. Answer C. Key: cylinder surface = two bases + lateral area.Given x + y = 9 and xy = 20, what is x2 + y2?
Correct answer: B
x2 + y2 = (x+y)2 - 2xy = 92 - 2×20 = 81 - 40 = 41. Answer B. Key: identity linking sum of squares to sum and product.In how many ways can 12 be written as a sum of three positive integers, where order does not matter?
Correct answer: C
Let a ≤ b ≤ c with a+b+c = 12. For a=1: 5 ways — (1,1,10),(1,2,9),(1,3,8),(1,4,7),(1,5,6); a=2: 4 ways — (2,2,8),(2,3,7),(2,4,6),(2,5,5); a=3: 2 ways — (3,3,6),(3,4,5); a=4: 1 way — (4,4,4). Total 12. Answer C. Key: ordered enumeration to avoid duplicates.Unified AMC 10 specs, consistent across years.
Papers A and B are comparable in difficulty and scope. Students may take either, or both — the higher score is used for awards and AIME qualification.
Four steps · Overview / Topics / Practice / Prep
Read the 2023 AMC 10 A/B overview and format specs to grasp the year’s style.
Review the key topics and weights, and locate your weak areas.
Complete the 25 mock questions (75-minute limit); submit for auto-grading and right/wrong marks.
Open the step-by-step solutions, learn from mistakes, and scan for full papers to push for AIME.
Prep tips and score-boosting advice for the 2023 AMC 10 A/B paper.
Finish all 25 within 75 minutes: pace the first 10 quickly and reserve at least 20 minutes for the last 5.
For quadratics, use Vieta and cross-multiplication instead of substituting roots one by one to save time.
In counting problems, enumerate by the leading or smallest element to stay complete and non-overlapping.
Open the solution for each error, note the key concept, and revisit similar problems to consolidate.
Aiming above 100? Lock down basics and medium questions, then tackle the last 5; scan for full papers and a tailored plan.
Frequent questions about the 2023 AMC 10 papers and answers.
Scan to add the advisor for 2023 AMC 10 A/B past papers, answer keys and solutions, plus a custom prep plan.
Note: Name + School + Grade + Year
