The 2014 AMC 10 Papers A and B carried a slightly higher share of algebra and number-theory divisibility, with factor-counting, handshake-counting, units-digit and cylinder-surface-area problems, stressing flexible formula use and modeling with counting. Scan for the full papers and answer keys.
2014 AMC 10 A/B paper overview and exam highlights.
The 2014 AMC 10 was held in November 2014 in two papers (A and B), with algebra and divisibility in number theory taking a slightly larger share.
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 2014 paper set more problems on divisor counts, handshake counting, units digits of powers, and cylinder surface area, stressing flexible use of formulas.
2014 AMC 10 main topic areas, weights and emphasis.
Evaluation, quadratics, word and speed problems; about 30%.
Rectangle perimeter, circle circumference, triangles and cylinder surface area; about 22%.
Divisibility, remainders, GCD and divisor counts; about 18%.
Coin outcomes, combinations and handshake counting; about 20%.
Classical probability and dice sums; about 10%.
2014 AMC 10-style mock questions (Basic · Medium · Hard) — submit to grade and view step-by-step solutions.
If x = 6, what is the value of x2 - 2x + 5?
Correct answer: C
Substitute x = 6: 62 - 2(6) + 5 = 36 - 12 + 5 = 29. Answer C. Key: direct substitution.The sum of three consecutive integers is 69. What is the largest?
Correct answer: D
Let the middle integer be n: 3n = 69, so n = 23 and the largest is 24. Answer D. Key: model consecutive integers by their middle.How many positive divisors does 180 have?
Correct answer: D
180 = 22×32×5, so the number of divisors is (2+1)(2+1)(1+1) = 18. Answer D. Key: divisor-count formula.A rectangle has length 9 and width 6. What is its perimeter?
Correct answer: B
Perimeter = 2(9 + 6) = 2×15 = 30. Answer B. (54 is the area — do not confuse.) Key: rectangle perimeter formula.An item is sold at 75% of its original price (25% off) for $60. What was the original price?
If x2 - 9x + 20 = 0, what is the sum of all possible values of x?
Correct answer: D
Factor: (x-4)(x-5) = 0, so x = 4 or 5, and the sum is 9. Answer D. Key: Vieta — sum of roots = -b/a = 9.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 16 people. How many different committees are possible?
Correct answer: B
C(16,2) = (16×15)/(2×1) = 120. Answer B. Key: combinations are order-independent, C(n,k) = n!/(k!(n-k)!).In a triangle, two angles are 41° and 49°. What is the third angle?
Correct answer: B
Triangle angles sum to 180°, so the third angle = 180 - 41 - 49 = 90°. Answer B. Key: triangle angle-sum theorem.The average of 7 numbers is 19. After removing one number, the average of the remaining 6 is 18. What was the removed number?
Correct answer: C
Sum of 7 = 7×19 = 133; sum of 6 = 6×18 = 108; the removed number = 133 - 108 = 25. Answer C. Key: average = sum / count.What is the units digit of 22014?
Correct answer: B
The units digit of powers of 2 cycles as 2, 4, 8, 6 (period 4). Since 2014 = 4×503 + 2, the remainder is 2, corresponding to the 2nd value, so the units digit is 4. Answer B. Key: cyclic units digits of powers.One worker can finish a job in 15 hours and another in 30 hours. How long does it take them together?
Correct answer: B
Rates: 1/15 + 1/30 = 2/30 + 1/30 = 3/30 = 1/10 per hour, so together they need 1 / (1/10) = 10 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: E
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 E. Key: enumerate ordered dice pairs.A car travels 120 km at 40 km/h and returns at 120 km/h. What is the average speed for the whole trip (km/h)?
Correct answer: C
Outbound time = 120/40 = 3 h; return = 120/120 = 1 h. Total distance 240 km over 4 h gives 240/4 = 60 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 6?
Correct answer: D
Divisible by both 4 and 6 means divisible by LCM(4,6) = 12. The multiples of 12 below 100 are 12, 24, 36, 48, 60, 72, 84, 96 — 8 of them. Answer D. Key: divisible by both ⟺ divisible by their LCM.At a party of 6 people, every pair shakes hands once. How many handshakes occur?
Correct answer: B
Each pair gives one handshake, so the count is C(6,2) = (6×5)/(2×1) = 15. Answer B. Key: handshake counting equals unordered pairing.A right cylinder has radius 7 and height 4. What is its total surface area? (in terms of π)
Correct answer: C
Surface area = 2πr2 + 2πrh = 2π(72) + 2π(7)(4) = 98π + 56π = 154π. Answer C. Key: cylinder surface = two bases + lateral area.Given x + y = 18 and xy = 77, what is x2 + y2?
Correct answer: A
x2 + y2 = (x+y)2 - 2xy = 182 - 2×77 = 324 - 154 = 170. Answer A. Key: identity linking sum of squares to sum and product.In how many ways can 21 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 = 21. For a=1: 10 ways — (1,1,19) to (1,10,10); a=2: 8 ways — (2,2,17) to (2,9,10); a=3: 7 ways — (3,3,15),(3,4,14),(3,5,13),(3,6,12),(3,7,11),(3,8,10),(3,9,9); a=4: 5 ways — (4,4,13),(4,5,12),(4,6,11),(4,7,10),(4,8,9); a=5: 4 ways — (5,5,11),(5,6,10),(5,7,9),(5,8,8); a=6: 2 ways — (6,6,9),(6,7,8); a=7: 1 way — (7,7,7). Total 37. 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 2014 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 2014 AMC 10 A/B paper.
Finish all 25 within 75 minutes; do the solvable ones first, then focus on the hard ones at the end.
After prime factoring, multiply (each exponent + 1) to get the divisor count.
Pairwise handshakes among n people equal C(n,2), the same as combinations.
Powers of 2, 3, 7 cycle with period 4; 9 with period 2; 5 and 6 are 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 2014 AMC 10 papers and answers.
Scan to add the advisor for 2014 AMC 10 A/B past papers, answer keys and solutions, plus a custom prep plan.
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