The 2021 AMC 10 Papers A and B stood out for difficulty in the algebra and combinatorics blocks, with algebraic-evaluation-and-equation, permutation-combination, divisibility and units-digit problems concentrated, demanding computational steadiness and back-half problem-solving. Scan for the full papers and answer keys.
2021 AMC 10 A/B paper overview and exam highlights.
The 2021 AMC 10 was held in November 2021 in two papers (A and B), with overall difficulty standing out in algebra and combinatorics.
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 (Paper A).
The 2021 paper concentrated on algebraic evaluation and equations, permutations and combinations, divisibility and units digits, demanding computational steadiness.
2021 AMC 10 main topic areas, weights and emphasis.
Evaluation, quadratics, speed and work-rate problems; about 32%.
Rectangle perimeter, circle area/circumference, triangles and cylinders; about 21%.
Divisibility, remainders, GCD and units digits of powers; about 16%.
Permutations, combinations and ordered counting; about 21%.
Classical probability and dice sums; about 10%.
2021 AMC 10-style mock questions (Basic · Medium · Hard) — submit to grade and view step-by-step solutions.
If x = 2, what is the value of x3 + 3x - 1?
Correct answer: C
Substitute x = 2: 23 + 3(2) - 1 = 8 + 6 - 1 = 13. Answer C. Key: direct substitution.The sum of five consecutive integers is 115. What is the largest?
Correct answer: D
Let the smallest be n: n+(n+1)+...+(n+4) = 5n+10 = 115, so 5n = 105, n = 21, and the largest is 25. Answer D. Key: model a consecutive run by its first term.How many positive divisors does 27 have?
Correct answer: B
27 = 33, so the number of divisors is 3+1 = 4 (namely 1,3,9,27). Answer B. Key: divisor count of a prime power.A rectangle has length 12 and width 5. What is its perimeter?
Correct answer: B
Perimeter = 2(12 + 5) = 2×17 = 34. Answer B. (60 is the area — do not confuse.) Key: rectangle perimeter formula.An item is marked up 10% above cost to a price of $55. What was the cost?
Correct answer: C
cost × 1.1 = 55, so cost = 55 / 1.1 = 50. Answer C. Key: price = cost × (1 + markup).A fair six-sided die is rolled once. What is the probability of rolling a number less than 3 (1 or 2)?
Correct answer: B
Outcomes less than 3 are 1 and 2 (2 of 6), so the probability is 2/6 = 1/3. Answer B. Key: classical probability = favorable / total.From 6 people, a president and a vice president are chosen (different people). How many ways?
Correct answer: C
President 6 choices, vice president 5, giving 6×5 = 30. Answer C. Key: ordered selection uses P(6,2).What is the remainder when 2021 is divided by 11?
Correct answer: D
11 × 183 = 2013, and 2021 - 2013 = 8, so the remainder is 8. Answer D. Key: division with remainder.If 7x - 9 = 19, what is x?
Correct answer: C
7x = 19 + 9 = 28, so x = 4. Answer C. Key: isolate the variable.A circle has radius 14. What is its area? (Use π ≈ 22/7)
Correct answer: D
Area = πr2 = (22/7)×142 = (22/7)×196 = 22×28 = 616. Answer D. Key: circle area formula.If x2 - 6x + 8 = 0, what is the sum of all possible values of x?
Correct answer: D
Factor: (x-2)(x-4) = 0, so x = 2 or 4, and the sum is 6. Answer D. Key: Vieta — sum of roots = -b/a = 6.What is the greatest common divisor of 36 and 60?
Correct answer: B
36 = 22×32 and 60 = 22×3×5, so GCD = 22×3 = 12. Answer B. Key: take the smallest power of each shared prime.A committee of 2 is chosen from 9 people. How many different committees are possible?
Correct answer: B
C(9,2) = (9×8)/(2×1) = 36. Answer B. Key: combinations are order-independent, C(n,k) = n!/(k!(n-k)!).In a triangle, two angles are 25° and 95°. What is the third angle?
Correct answer: C
Triangle angles sum to 180°, so the third angle = 180 - 25 - 95 = 60°. Answer C. Key: triangle angle-sum theorem.The average of 7 numbers is 12. After removing one number, the average of the remaining 6 is 11. What was the removed number?
Correct answer: C
Sum of 7 = 7×12 = 84; sum of 6 = 6×11 = 66; the removed number = 84 - 66 = 18. Answer C. Key: average = sum / count.What is the units digit of 82021?
Correct answer: D
The units digit of powers of 8 cycles as 8, 4, 2, 6 (period 4). Since 2021 = 4×505 + 1, the remainder is 1, corresponding to the 1st value, so the units digit is 8. Answer D. Key: cyclic units digits of powers.One worker can finish a job in 5 hours and another in 20 hours. How long does it take them together?
Correct answer: C
Rates: 1/5 + 1/20 = 4/20 + 1/20 = 5/20 = 1/4 per hour, so together they need 1 / (1/4) = 4 hours. Answer C. Key: combined rate = sum of individual rates.A square and an equilateral triangle have the same perimeter. If the triangle side is 4, what is the square side?
Correct answer: C
Triangle perimeter = 3×4 = 12, so the square side = 12 / 4 = 3. Answer C. Key: equal perimeters split by side count.Two fair six-sided dice are rolled. What is the probability the sum is 10?
Correct answer: A
Sums to 10: (4,6),(5,5),(6,4) — 3 outcomes out of 36, so 3/36 = 1/12. Answer A. Key: enumerate ordered dice pairs.A car travels 300 km at 60 km/h and returns at 100 km/h. What is the average speed for the whole trip (km/h)?
Correct answer: C
Outbound time = 300/60 = 5 h; return = 300/100 = 3 h. Total distance 600 km over 8 h gives 600/8 = 75 km/h. Answer C. Key: average speed = total distance / total time, not the average of speeds.How many positive integers less than 100 are divisible by both 4 and 10?
Correct answer: C
Divisible by both 4 and 10 means divisible by LCM(4,10) = 20. The multiples of 20 below 100 are 20, 40, 60, 80 — 4 of them. Answer C. Key: divisible by both ⟺ divisible by their LCM.From 7 distinct objects, 3 are chosen. How many selections are possible?
Correct answer: C
C(7,3) = (7×6×5)/(3×2×1) = 210/6 = 35. Answer C. Key: combinations are order-independent, C(n,k) = n!/(k!(n-k)!).A right cylinder has radius 6 and height 3. What is its total surface area? (in terms of π)
Correct answer: C
Surface area = 2πr2 + 2πrh = 2π(62) + 2π(6)(3) = 72π + 36π = 108π. Answer C. Key: cylinder surface = two bases + lateral area.Given x + y = 11 and xy = 30, what is x2 + y2?
Correct answer: C
x2 + y2 = (x+y)2 - 2xy = 112 - 2×30 = 121 - 60 = 61. Answer C. Key: identity linking sum of squares to sum and product.In how many ways can 14 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 = 14. For a=1: 6 ways — (1,1,12),(1,2,11),(1,3,10),(1,4,9),(1,5,8),(1,6,7); a=2: 5 ways — (2,2,10),(2,3,9),(2,4,8),(2,5,7),(2,6,6); a=3: 3 ways — (3,3,8),(3,4,7),(3,5,6); a=4: 2 ways — (4,4,6),(4,5,5). Total 16. 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 2021 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 2021 AMC 10 A/B paper.
Finish all 25 within 75 minutes; when stuck, skip ahead so the solvable ones score fully first.
Write substitution and equation-solving step by step to avoid sign and arithmetic slips.
Decide ordered vs. unordered and repeatable vs. not before choosing a permutation or combination.
The units-digit cycles of 8 and 9 have period 4 and 2; locate the result by the remainder.
Aiming above 100? Lock the basics and medium questions, then take the last 5 step by step; scan for full papers and a plan.
Frequent questions about the 2021 AMC 10 papers and answers.
Scan to add the advisor for 2021 AMC 10 A/B past papers, answer keys and solutions, plus a custom prep plan.
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