The 2009 AMC 10 Papers A and B emphasized algebraic evaluation and number-theory divisibility, with average-speed, coin-counting, units-digit and combinatorial-counting problems concentrated in the back half, raising the demand on accurate setup and rigorous enumeration. Scan for the full papers and answer keys.
2009 AMC 10 A/B paper overview and exam highlights.
The 2009 AMC 10 was held in November 2009 in two papers (A and B), with algebraic evaluation and divisibility in number theory 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 2009 paper set more problems on average speed, coin-outcome counting, units digits of powers, and combinatorial counting, combining formulas with careful reading.
2009 AMC 10 main topic areas, weights and emphasis.
Evaluation, quadratics, word and speed problems; about 31%.
Rectangle perimeter, triangle area, cylinder surface area; about 22%.
Divisibility, remainders, GCD and divisor counts; about 17%.
Coin outcomes, combinations and ordered counting; about 20%.
Classical probability and dice sums; about 10%.
2009 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 - 4x + 5?
Correct answer: A
Substitute x = 6: 62 - 4(6) + 5 = 36 - 24 + 5 = 17. Answer A. Key: direct substitution.The sum of three consecutive integers is 102. What is the largest?
Correct answer: B
Let the middle integer be n: 3n = 102, so n = 34 and the largest is 35. Answer B. Key: model consecutive integers by their middle.How many positive divisors does 200 have?
Correct answer: C
200 = 23×52, so the number of divisors is (3+1)(2+1) = 12. Answer C. Key: divisor-count formula.A rectangle has length 9 and width 6. What is its perimeter?
Correct answer: D
Perimeter = 2(9 + 6) = 2×15 = 30. Answer D. (54 is the area — do not confuse.) Key: rectangle perimeter formula.An item is marked up 25% above cost to a price of
A fair six-sided die is rolled once. What is the probability of rolling a prime number (2, 3, or 5)?
Correct answer: A
Primes are 2, 3, 5 (3 of 6), so the probability is 3/6 = 1/2. Answer A. Key: classical probability = favorable / total.Seven coins are tossed at once. How many heads/tails outcomes are possible?
Correct answer: B
Each coin has 2 outcomes, so 7 coins give 27 = 128. Answer B. Key: independent outcomes multiply.What is the remainder when 2009 is divided by 7?
Correct answer: C
7 × 287 = 2009, so it divides exactly and the remainder is 0. Answer C. Key: division with remainder — watch the exact-division case.If 4x - 6 = 18, what is x?
Correct answer: D
4x = 18 + 6 = 24, so x = 6. Answer D. Key: isolate the variable.A triangle has base 9 and height 8. What is its area?
Correct answer: E
Area = ½×9×8 = 72/2 = 36. Answer E. Key: triangle area formula.If x2 - 15x + 56 = 0, what is the sum of all possible values of x?
Correct answer: A
Factor: (x-7)(x-8) = 0, so x = 7 or 8, and the sum is 15. Answer A. Key: Vieta — sum of roots = -b/a = 15.What is the greatest common divisor of 42 and 63?
Correct answer: B
42 = 2×3×7 and 63 = 32×7, so GCD = 3×7 = 21. Answer B. Key: take the smallest power of each shared prime.A committee of 2 is chosen from 22 people. How many different committees are possible?
Correct answer: C
C(22,2) = (22×21)/(2×1) = 231. Answer C. Key: combinations are order-independent, C(n,k) = n!/(k!(n-k)!).In a triangle, two angles are 38° and 52°. What is the third angle?
Correct answer: D
Triangle angles sum to 180°, so the third angle = 180 - 38 - 52 = 90°. Answer D. Key: triangle angle-sum theorem.The average of 9 numbers is 11. After removing one number, the average of the remaining 8 is 10. What was the removed number?
Correct answer: E
Sum of 9 = 9×11 = 99; sum of 8 = 8×10 = 80; the removed number = 99 - 80 = 19. Answer E. Key: average = sum / count.What is the units digit of 32009?
Correct answer: A
The units digit of powers of 3 cycles as 3, 9, 7, 1 (period 4). Since 2009 = 4×502 + 1, the remainder is 1, corresponding to the 1st value, so the units digit is 3. 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: B
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 B. Key: combined rate = sum of individual rates.A square and an equilateral triangle have the same perimeter. If the triangle side is 36, what is the square side?
Correct answer: C
Triangle perimeter = 3×36 = 108, so the square side = 108 / 4 = 27. Answer C. Key: equal perimeters split by side count.Two fair six-sided dice are rolled. What is the probability the sum is 4?
Correct answer: D
Sums to 4: (1,3),(2,2),(3,1) — 3 outcomes out of 36, so 3/36 = 1/12. Answer D. Key: enumerate ordered dice pairs.A car travels 120 km at 40 km/h and returns at 60 km/h. What is the average speed for the whole trip (km/h)?
Correct answer: E
Outbound time = 120/40 = 3 h; return = 120/60 = 2 h. Total distance 240 km over 5 h gives 240/5 = 48 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 5 and 8?
Correct answer: A
Divisible by both 5 and 8 means divisible by LCM(5,8) = 40. The multiples of 40 below 100 are 40, 80 — 2 of them. Answer A. Key: divisible by both ⟺ divisible by their LCM.A committee of 3 is chosen from 10 people. How many different committees are possible?
Correct answer: B
C(10,3) = (10×9×8)/(3×2×1) = 720/6 = 120. Answer B. Key: combinations are order-independent, C(n,k) = n!/(k!(n-k)!).A right cylinder has radius 4 and height 4. What is its total surface area? (in terms of π)
Correct answer: C
Surface area = 2πr2 + 2πrh = 2π(42) + 2π(4)(4) = 32π + 32π = 64π. Answer C. Key: cylinder surface = two bases + lateral area.Given x + y = 24 and xy = 119, what is x2 + y2?
Correct answer: D
x2 + y2 = (x+y)2 - 2xy = 242 - 2×119 = 576 - 238 = 338. Answer D. Key: identity linking sum of squares to sum and product.In how many ways can 26 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 = 26. For a=1: 12 ways — (1,1,24) to (1,12,13); a=2: 11 ways — (2,2,22) to (2,12,12); a=3: 9 ways — (3,3,20) to (3,11,12); a=4: 8 ways — (4,4,18) to (4,11,11); a=5: 6 ways — (5,5,16) to (5,10,11); a=6: 5 ways — (6,6,14) to (6,10,10); a=7: 3 ways — (7,7,12) to (7,9,10); a=8: 2 ways — (8,8,10) to (8,9,9). Total 56. 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 2009 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 2009 AMC 10 A/B paper.
Finish all 25 within 75 minutes; skip when stuck, bank the solvable ones, and return to the hard ones.
Average speed = total distance / total time — never the arithmetic mean of the two speeds.
Divisible by both numbers means divisible by their LCM; use division with remainder for quotients.
Ordered uses P(n,k); unordered uses C(n,k); check order and repeatability.
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 2009 AMC 10 papers and answers.
Scan to add the advisor for 2009 AMC 10 A/B past papers, answer keys and solutions, plus a custom prep plan.
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