The 2005 AMC 10 Papers A and B continued the algebraic-evaluation and number-theory-divisibility thrust, with staple problems on factor-counting, coin outcomes, units digits and average speed testing fluency with core formulas and conceptual discrimination. Scan for the full papers and answer keys.
2005 AMC 10 A/B paper overview and exam highlights.
The 2005 AMC 10 was held in November 2005 in two papers (A and B), fairly balanced between algebraic evaluation and divisibility in number theory.
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 2005 paper set more problems on divisor counts, coin outcomes, units digits of powers, and average speed, combining formulas with careful reading.
2005 AMC 10 main topic areas, weights and emphasis.
Evaluation, quadratics, word and speed problems; about 28%.
Rectangle perimeter, triangle area, cylinder surface area; about 23%.
Divisibility, remainders, GCD and divisor counts; about 19%.
Coin outcomes, combinations and ordered counting; about 20%.
Classical probability and dice sums; about 10%.
2005 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 3x2 + 4x - 2?
Correct answer: A
Substitute x = 2: 3(2)2 + 4(2) - 2 = 3×4 + 8 - 2 = 12 + 8 - 2 = 18. Answer A. Key: direct substitution.The sum of three consecutive integers is 126. What is the largest?
Correct answer: B
Let the middle integer be n: 3n = 126, so n = 42 and the largest is 43. Answer B. Key: model consecutive integers by their middle.How many positive divisors does 144 have?
Correct answer: C
144 = 24×32, so the number of divisors is (4+1)(2+1) = 15. Answer C. Key: divisor-count formula.A rectangle has length 10 and width 4. What is its perimeter?
Correct answer: D
Perimeter = 2(10 + 4) = 2×14 = 28. Answer D. (40 is the area — do not confuse.) Key: rectangle perimeter formula.An item is marked up 30% above cost to a price of $78. What was the cost?
Correct answer: E
cost × 1.3 = 78, so cost = 78 / 1.3 = 60. Answer E. Key: price = cost × (1 + markup).A fair six-sided die is rolled once. What is the probability of rolling at least 5 (5 or 6)?
Correct answer: A
Values at least 5 are 5 and 6 (2 of 6), so the probability is 2/6 = 1/3. Answer A. Key: classical probability = favorable / total.Four coins are tossed at once. How many heads/tails outcomes are possible?
Correct answer: B
Each coin has 2 outcomes, so 4 coins give 24 = 16. Answer B. Key: independent outcomes multiply.What is the remainder when 2005 is divided by 7?
Correct answer: C
7 × 286 = 2002, and 2005 - 2002 = 3, so the remainder is 3. Answer C. Key: division with remainder.If 6x + 5 = 41, what is x?
Correct answer: D
6x = 41 - 5 = 36, so x = 6. Answer D. Key: isolate the variable.A triangle has base 11 and height 6. What is its area?
Correct answer: E
Area = ½×11×6 = 66/2 = 33. Answer E. Key: triangle area formula.If x2 - 14x + 48 = 0, what is the sum of all possible values of x?
Correct answer: A
Factor: (x-6)(x-8) = 0, so x = 6 or 8, and the sum is 14. Answer A. Key: Vieta — sum of roots = -b/a = 14.What is the greatest common divisor of 60 and 84?
Correct answer: B
60 = 22×3×5 and 84 = 22×3×7, so GCD = 22×3 = 12. Answer B. 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: C
C(16,2) = (16×15)/(2×1) = 120. Answer C. Key: combinations are order-independent, C(n,k) = n!/(k!(n-k)!).In a triangle, two angles are 35° and 55°. What is the third angle?
Correct answer: D
Triangle angles sum to 180°, so the third angle = 180 - 35 - 55 = 90°. Answer D. Key: triangle angle-sum theorem.The average of 6 numbers is 20. After removing one number, the average of the remaining 5 is 18. What was the removed number?
Correct answer: E
Sum of 6 = 6×20 = 120; sum of 5 = 5×18 = 90; the removed number = 120 - 90 = 30. Answer E. Key: average = sum / count.What is the units digit of 72005?
Correct answer: A
The units digit of powers of 7 cycles as 7, 9, 3, 1 (period 4). Since 2005 = 4×501 + 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 10 hours and another in 40 hours. How long does it take them together?
Correct answer: B
Rates: 1/10 + 1/40 = 4/40 + 1/40 = 5/40 = 1/8 per hour, so together they need 1 / (1/8) = 8 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 48, what is the square side?
Correct answer: C
Triangle perimeter = 3×48 = 144, so the square side = 144 / 4 = 36. 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: D
Sums to 10: (4,6),(5,5),(6,4) — 3 outcomes out of 36, so 3/36 = 1/12. Answer D. Key: enumerate ordered dice pairs.A car travels 180 km at 30 km/h and returns at 90 km/h. What is the average speed for the whole trip (km/h)?
Correct answer: E
Outbound time = 180/30 = 6 h; return = 180/90 = 2 h. Total distance 360 km over 8 h gives 360/8 = 45 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 4 and 10?
Correct answer: A
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 A. Key: divisible by both ⟺ divisible by their LCM.A committee of 3 is chosen from 9 people. How many different committees are possible?
Correct answer: B
C(9,3) = (9×8×7)/(3×2×1) = 504/6 = 84. Answer B. Key: combinations are order-independent, C(n,k) = n!/(k!(n-k)!).A right cylinder has radius 6 and height 2. What is its total surface area? (in terms of π)
Correct answer: C
Surface area = 2πr2 + 2πrh = 2π(62) + 2π(6)(2) = 72π + 24π = 96π. Answer C. Key: cylinder surface = two bases + lateral area.Given x + y = 29 and xy = 210, what is x2 + y2?
Correct answer: D
x2 + y2 = (x+y)2 - 2xy = 292 - 2×210 = 841 - 420 = 421. Answer D. Key: identity linking sum of squares to sum and product.In how many ways can 30 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 = 30. For a=1: 14 ways — (1,1,30) to (1,14,15); a=2: 13 ways — (2,2,26) to (2,14,14); a=3: 11 ways — (3,3,24) to (3,13,14); a=4: 10 ways — (4,4,22) to (4,13,13); a=5: 8 ways — (5,5,20) to (5,12,13); a=6: 7 ways — (6,6,18) to (6,12,12); a=7: 5 ways — (7,7,16) to (7,11,12); a=8: 4 ways — (8,8,14) to (8,11,11); a=9: 2 ways — (9,9,12) to (9,10,11); a=10: 1 way — (10,10,10). Total 75. 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 2005 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 2005 AMC 10 A/B paper.
Finish all 25 within 75 minutes; skip when stuck, bank the solvable ones, and return to the hard ones.
Factor N into prime powers; the divisor count is the product of (exponent + 1).
Units digits cycle by the base; locate the result via the exponent mod the period.
Average speed = total distance / total time — never the arithmetic mean of the two speeds.
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 2005 AMC 10 papers and answers.
Scan to add the advisor for 2005 AMC 10 A/B past papers, answer keys and solutions, plus a custom prep plan.
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