The 2013 AMC 10 Papers A and B balanced algebra with combinatorial counting, with factor-counting, nonempty-proper-subset counting, units-digit and average-speed problems across the back half, emphasizing conceptual discrimination and counting principles. Scan for the full papers and answer keys.
2013 AMC 10 A/B paper overview and exam highlights.
The 2013 AMC 10 was held in November 2013 in two papers (A and B), fairly balanced between algebra and combinatorial counting.
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 2013 paper set more problems on divisor counts, nonempty proper subset counting, units digits of powers, and average speed, stressing conceptual distinctions.
2013 AMC 10 main topic areas, weights and emphasis.
Evaluation, quadratics, word and speed problems; about 29%.
Rectangle perimeter, triangle area, circles and cylinders; about 23%.
Divisibility, remainders, GCD and divisor counts; about 17%.
Coin outcomes, combinations and nonempty proper subset counting; about 21%.
Classical probability and dice sums; about 10%.
2013 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: C
Substitute x = 5: 2(5)2 - 3(5) + 4 = 50 - 15 + 4 = 39. Answer C. Key: direct substitution.The sum of three consecutive integers is 105. What is the largest?
Correct answer: D
Let the middle integer be n: 3n = 105, so n = 35 and the largest is 36. Answer D. Key: model consecutive integers by their middle.How many positive divisors does 216 have?
Correct answer: C
216 = 23×33, so the number of divisors is (3+1)(3+1) = 16. Answer C. Key: divisor-count formula.An equilateral triangle has side length 9. What is its perimeter?
Correct answer: B
All three sides are equal, so the perimeter = 3×9 = 27. Answer B. Key: equilateral triangle perimeter.An item is marked up 20% above cost to a price of
Two fair six-sided dice are rolled. What is the probability of rolling doubles (equal numbers)?
Correct answer: A
Doubles: (1,1),(2,2),(3,3),(4,4),(5,5),(6,6) — 6 outcomes out of 36, so 6/36 = 1/6. Answer A. Key: enumerate ordered dice pairs.In how many ways can 6 different books be arranged in a row?
Correct answer: D
Permutations of 6 distinct items: 6! = 720. Answer D. Key: n distinct items have n! arrangements.What is the remainder when 2013 is divided by 7?
Correct answer: E
7 × 287 = 2009, and 2013 - 2009 = 4, so the remainder is 4. Answer E. Key: division with remainder.If 3x - 5 = 25, what is x?
Correct answer: C
3x = 25 + 5 = 30, so x = 10. Answer C. Key: isolate the variable.A triangle has base 14 and height 5. What is its area?
Correct answer: B
Area = ½×14×5 = 70/2 = 35. Answer B. Key: triangle area formula.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 63 and 84?
Correct answer: C
63 = 32×7 and 84 = 22×3×7, so GCD = 3×7 = 21. Answer C. 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: B
C(18,2) = (18×17)/(2×1) = 153. Answer B. Key: combinations are order-independent, C(n,k) = n!/(k!(n-k)!).In a triangle, two angles are 38° and 62°. What is the third angle?
Correct answer: B
Triangle angles sum to 180°, so the third angle = 180 - 38 - 62 = 80°. Answer B. Key: triangle angle-sum theorem.The average of 7 numbers is 14. After removing one number, the average of the remaining 6 is 11. What was the removed number?
Correct answer: C
Sum of 7 = 7×14 = 98; sum of 6 = 6×11 = 66; the removed number = 98 - 66 = 32. Answer C. Key: average = sum / count.What is the units digit of 72013?
Correct answer: A
The units digit of powers of 7 cycles as 7, 9, 3, 1 (period 4). Since 2013 = 4×503 + 1, the remainder is 1, corresponding to the 1st value, so the units digit is 7. Answer A. Key: cyclic units digits of powers.One worker can finish a job in 6 hours and another in 30 hours. How long does it take them together?
Correct answer: C
Rates: 1/6 + 1/30 = 5/30 + 1/30 = 6/30 = 1/5 per hour, so together they need 1 / (1/5) = 5 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 20, what is the square side?
Correct answer: C
Triangle perimeter = 3×20 = 60, so the square side = 60 / 4 = 15. Answer C. Key: equal perimeters split by side count.Two fair six-sided dice are rolled. What is the probability the sum is 2?
Correct answer: A
The only sum-2 outcome is (1,1) — 1 out of 36, so 1/36. Answer A. Key: enumerate ordered dice pairs.A car travels 120 km at 30 km/h and returns at 120 km/h. What is the average speed for the whole trip (km/h)?
Correct answer: C
Outbound time = 120/30 = 4 h; return = 120/120 = 1 h. Total distance 240 km over 5 h gives 240/5 = 48 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 8?
Correct answer: C
Divisible by both 4 and 8 means divisible by LCM(4,8) = 8. The multiples of 8 below 100 are 8, 16, …, 96 — 12 of them. Answer C. Key: divisible by both ⟺ divisible by their LCM.A set has 6 elements. How many nonempty proper subsets does it have (excluding the empty set and the set itself)?
Correct answer: B
An n-element set has 2n subsets, so nonempty proper subsets = 26 - 2 = 64 - 2 = 62. Answer B. Key: subset count minus the empty set and itself.A right cylinder has radius 3 and height 4. What is its total surface area? (in terms of π)
Correct answer: C
Surface area = 2πr2 + 2πrh = 2π(32) + 2π(3)(4) = 18π + 24π = 42π. Answer C. Key: cylinder surface = two bases + lateral area.Given x + y = 20 and xy = 91, what is x2 + y2?
Correct answer: A
x2 + y2 = (x+y)2 - 2xy = 202 - 2×91 = 400 - 182 = 218. Answer A. Key: identity linking sum of squares to sum and product.In how many ways can 22 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 = 22. For a=1: 10 ways — (1,1,20) to (1,10,11); a=2: 9 ways — (2,2,18) to (2,10,10); a=3: 7 ways — (3,3,16),(3,4,15),(3,5,14),(3,6,13),(3,7,12),(3,8,11),(3,9,10); a=4: 6 ways — (4,4,14),(4,5,13),(4,6,12),(4,7,11),(4,8,10),(4,9,9); a=5: 4 ways — (5,5,12),(5,6,11),(5,7,10),(5,8,9); a=6: 3 ways — (6,6,10),(6,7,9),(6,8,8); a=7: 1 way — (7,7,8). Total 40. 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 2013 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 2013 AMC 10 A/B paper.
Finish all 25 within 75 minutes; budget time well and save the hard ones for last.
An n-element set has 2n subsets; nonempty proper subsets number 2n - 2.
Divisible by both numbers means divisible by their LCM — find the LCM, then count multiples.
Memorize the cycles of 2, 3, 7, 8 (period 4), 9 (period 2), and 5/6 (fixed).
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 2013 AMC 10 papers and answers.
Scan to add the advisor for 2013 AMC 10 A/B past papers, answer keys and solutions, plus a custom prep plan.
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