The 2010 AMC 10 Papers A and B carried a slightly higher share of algebraic evaluation and geometric counting, with regular-polygon perimeter, coin-outcome, units-digit and average-speed staples; geometry-computation and counting crossover items emphasized diagramming speed. Scan for the full papers and answer keys.
2010 AMC 10 A/B paper overview and exam highlights.
The 2010 AMC 10 was held in November 2010 in two papers (A and B), with algebraic evaluation and geometric counting standing out.
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 2010 paper set more problems on regular-polygon perimeter, coin outcomes, units digits of powers, and average speed, combining formulas with careful reading.
2010 AMC 10 main topic areas, weights and emphasis.
Evaluation, quadratics, word and speed problems; about 28%.
Regular hexagons, rectangle perimeter, circles and cylinder surface area; about 24%.
Divisibility, remainders, GCD and divisor counts; about 18%.
Coin outcomes, permutations and combinations; about 20%.
Classical probability and dice sums; about 10%.
2010 AMC 10-style mock questions (Basic · Medium · Hard) — submit to grade and view step-by-step solutions.
If x = 3, what is the value of 4x2 - x + 2?
Correct answer: A
Substitute x = 3: 4(3)2 - 3 + 2 = 4×9 - 3 + 2 = 36 - 3 + 2 = 35. Answer A. Key: direct substitution.The sum of three consecutive integers is 96. What is the largest?
Correct answer: B
Let the middle integer be n: 3n = 96, so n = 32 and the largest is 33. Answer B. Key: model consecutive integers by their middle.How many positive divisors does 270 have?
Correct answer: C
270 = 2×33×5, so the number of divisors is (1+1)(3+1)(1+1) = 16. Answer C. Key: divisor-count formula.A regular hexagon has side length 6. What is its perimeter?
Correct answer: D
All six sides of a regular hexagon are equal, so the perimeter = 6×6 = 36. Answer D. Key: regular-polygon perimeter.An item is marked up 40% above cost to a price of $98. What was the cost?
Correct answer: E
cost × 1.4 = 98, so cost = 98 / 1.4 = 70. Answer E. Key: price = cost × (1 + markup).A fair six-sided die is rolled once. What is the probability of rolling a divisor of 6 (1, 2, 3, or 6)?
Correct answer: A
Divisors of 6 are 1, 2, 3, 6 (4 of 6), so the probability is 4/6 = 2/3. Answer A. Key: classical probability = favorable / total.Six coins are tossed at once. How many heads/tails outcomes are possible?
Correct answer: B
Each coin has 2 outcomes, so 6 coins give 26 = 64. Answer B. Key: independent outcomes multiply.What is the remainder when 2010 is divided by 11?
Correct answer: C
11 × 182 = 2002, and 2010 - 2002 = 8, so the remainder is 8. Answer C. Key: division with remainder.If 3x + 9 = 30, what is x?
Correct answer: D
3x = 30 - 9 = 21, so x = 7. Answer D. Key: isolate the variable.A triangle has base 13 and height 6. What is its area?
Correct answer: E
Area = ½×13×6 = 78/2 = 39. Answer E. Key: triangle area formula.If x2 - 12x + 27 = 0, what is the sum of all possible values of x?
Correct answer: A
Factor: (x-3)(x-9) = 0, so x = 3 or 9, and the sum is 12. Answer A. Key: Vieta — sum of roots = -b/a = 12.What is the greatest common divisor of 56 and 84?
Correct answer: B
56 = 23×7 and 84 = 22×3×7, so GCD = 22×7 = 28. Answer B. Key: take the smallest power of each shared prime.A committee of 2 is chosen from 18 people. How many different committees are possible?
Correct answer: C
C(18,2) = (18×17)/(2×1) = 153. Answer C. Key: combinations are order-independent, C(n,k) = n!/(k!(n-k)!).In a triangle, two angles are 41° and 79°. What is the third angle?
Correct answer: D
Triangle angles sum to 180°, so the third angle = 180 - 41 - 79 = 60°. Answer D. Key: triangle angle-sum theorem.The average of 6 numbers is 15. After removing one number, the average of the remaining 5 is 12. What was the removed number?
Correct answer: E
Sum of 6 = 6×15 = 90; sum of 5 = 5×12 = 60; the removed number = 90 - 60 = 30. Answer E. Key: average = sum / count.What is the units digit of 42010?
Correct answer: A
The units digit of powers of 4 cycles as 4, 6 (period 2). Since 2010 = 2×1005, the remainder is 0, corresponding to the 2nd value, so the units digit is 6. Answer A. 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: B
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 B. Key: combined rate = sum of individual rates.A square and an equilateral triangle have the same perimeter. If the triangle side is 56, what is the square side?
Correct answer: C
Triangle perimeter = 3×56 = 168, so the square side = 168 / 4 = 42. Answer C. Key: equal perimeters split by side count.Two fair six-sided dice are rolled. What is the probability the sum is 7?
Correct answer: D
Sums to 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) — 6 outcomes out of 36, so 6/36 = 1/6. Answer D. Key: enumerate ordered dice pairs.A car travels 60 km at 30 km/h and returns at 60 km/h. What is the average speed for the whole trip (km/h)?
Correct answer: E
Outbound time = 60/30 = 2 h; return = 60/60 = 1 h. Total distance 120 km over 3 h gives 120/3 = 40 km/h. Answer E. Key: average speed = total distance / total time, not the average of speeds.How many positive integers less than 100 are divisible by both 6 and 15?
Correct answer: A
Divisible by both 6 and 15 means divisible by LCM(6,15) = 30. The multiples of 30 below 100 are 30, 60, 90 — 3 of them. Answer A. Key: divisible by both ⟺ divisible by their LCM.From 5 different books, 2 are chosen and arranged in order on a shelf. How many arrangements are possible?
Correct answer: B
Permutation P(5,2) = 5×4 = 20. Answer B. Key: ordered selection uses permutations, P(n,k) = n!/(n-k)!.A right cylinder has radius 4 and height 7. What is its total surface area? (in terms of π)
Correct answer: C
Surface area = 2πr2 + 2πrh = 2π(42) + 2π(4)(7) = 32π + 56π = 88π. Answer C. Key: cylinder surface = two bases + lateral area.Given x + y = 22 and xy = 117, what is x2 + y2?
Correct answer: D
x2 + y2 = (x+y)2 - 2xy = 222 - 2×117 = 484 - 234 = 250. Answer D. Key: identity linking sum of squares to sum and product.In how many ways can 25 be written as a sum of three positive integers, where order does not matter?
Correct answer: E
Let a ≤ b ≤ c with a+b+c = 25. For a=1: 12 ways — (1,1,23) to (1,12,12); a=2: 10 ways — (2,2,21) to (2,11,12); a=3: 9 ways — (3,3,19) to (3,11,11); a=4: 7 ways — (4,4,17) to (4,10,11); a=5: 6 ways — (5,5,15) to (5,10,10); a=6: 4 ways — (6,6,13) to (6,9,10); a=7: 3 ways — (7,7,11) to (7,9,9); a=8: 1 way — (8,8,9). Total 52. Answer E. 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 2010 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 2010 AMC 10 A/B paper.
Finish all 25 within 75 minutes; skip when stuck, bank the solvable ones, and return to the hard ones.
A regular n-gon has perimeter n × side; hexagons and triangles appear often.
Ordered uses P(n,k); unordered uses C(n,k); check order and repeatability.
Units digits cycle by the base; locate the result via the exponent mod the period.
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 2010 AMC 10 papers and answers.
Scan to add the advisor for 2010 AMC 10 A/B past papers, answer keys and solutions, plus a custom prep plan.
Note: Name + School + Grade + Year
