The 2000 AMC 10, held in November 2000, emphasized algebraic evaluation and number-theory divisibility, with factor-counting, coin-outcome and units-digit problems featuring prominently in the basics. Scan for the full paper and answer key.
2000 AMC 10 paper overview and exam highlights.
The 2000 AMC 10 was held in November 2000, 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.
The 2000 paper set more problems on divisor counts, coin outcomes, units digits of powers, and average speed, combining formulas with careful reading.
2000 AMC 10 main topic areas, weights and emphasis.
Evaluation, quadratics, word and speed problems; about 30%.
Rectangle perimeter, triangle area, cylinder surface area; about 22%.
Divisibility, remainders, GCD and divisor counts; about 18%.
Coin outcomes, combinations and ordered counting; about 20%.
Classical probability and dice sums; about 10%.
2000 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 - x + 3?
Correct answer: A
Substitute x = 5: 2(5)2 - 5 + 3 = 2×25 - 5 + 3 = 50 - 5 + 3 = 48. Answer A. Key: direct substitution.The sum of three consecutive integers is 144. What is the largest?
Correct answer: B
Let the middle integer be n: 3n = 144, so n = 48 and the largest is 49. Answer B. Key: model consecutive integers by their middle.How many positive divisors does 80 have?
Correct answer: C
80 = 24×5, so the number of divisors is (4+1)(1+1) = 10. Answer C. Key: divisor-count formula.A rectangle has length 14 and width 2. What is its perimeter?
Correct answer: D
Perimeter = 2(14 + 2) = 2×16 = 32. Answer D. (28 is length + width — do not confuse.) Key: rectangle perimeter formula.An item is marked up 25% above cost to a price of $50. What was the cost?
0Correct answer: E
cost × 1.25 = 50, so cost = 50 / 1.25 = 40. Answer E. Key: price = cost × (1 + markup).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.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 2000 is divided by 9?
Correct answer: C
9 × 222 = 1998, and 2000 - 1998 = 2, so the remainder is 2. Answer C. Key: division with remainder.If 6x + 1 = 31, what is x?
Correct answer: D
6x = 31 - 1 = 30, so x = 5. Answer D. Key: isolate the variable.A triangle has base 17 and height 4. What is its area?
Correct answer: E
Area = ½×17×4 = 68/2 = 34. Answer E. Key: triangle area formula.If x2 - 8x + 12 = 0, what is the sum of all possible values of x?
Correct answer: A
Factor: (x-2)(x-6) = 0, so x = 2 or 6, and the sum is 8. Answer A. Key: Vieta — sum of roots = -b/a = 8.What is the greatest common divisor of 40 and 60?
Correct answer: B
40 = 23×5 and 60 = 22×3×5, so GCD = 22×5 = 20. Answer B. Key: take the smallest power of each shared prime.A committee of 2 is chosen from 31 people. How many different committees are possible?
Correct answer: C
C(31,2) = (31×30)/(2×1) = 465. Answer C. Key: combinations are order-independent, C(n,k) = n!/(k!(n-k)!).In a triangle, two angles are 42° and 68°. What is the third angle?
Correct answer: D
Triangle angles sum to 180°, so the third angle = 180 - 42 - 68 = 70°. Answer D. Key: triangle angle-sum theorem.The average of 9 numbers is 16. After removing one number, the average of the remaining 8 is 15. What was the removed number?
Correct answer: E
Sum of 9 = 9×16 = 144; sum of 8 = 8×15 = 120; the removed number = 144 - 120 = 24. Answer E. Key: average = sum / count.What is the units digit of 22000?
Correct answer: A
The units digit of powers of 2 cycles as 2, 4, 8, 6 (period 4). Since 2000 = 4×500, the remainder is 0, corresponding to the 4th value, so the units digit is 6. Answer A. Key: cyclic units digits of powers.One worker can finish a job in 8 hours and another also in 8 hours. How long does it take them together?
Correct answer: B
Rates: 1/8 + 1/8 = 2/8 = 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 96, what is the square side?
Correct answer: C
Triangle perimeter = 3×96 = 288, so the square side = 288 / 4 = 72. Answer C. Key: equal perimeters split by side count.Two fair six-sided dice are rolled. What is the probability the sum is 8?
Correct answer: D
Sums to 8: (2,6),(3,5),(4,4),(5,3),(6,2) — 5 outcomes out of 36, so 5/36. 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 7 and 14?
Correct answer: A
Divisible by both 7 and 14 means divisible by LCM(7,14) = 14. The multiples of 14 below 100 are 14, 28, 42, 56, 70, 84, 98 — 7 of them. Answer A. Key: divisible by both ⟺ divisible by their LCM.A committee of 3 is chosen from 15 people. How many different committees are possible?
Correct answer: B
C(15,3) = (15×14×13)/(3×2×1) = 2730/6 = 455. Answer B. Key: combinations are order-independent, C(n,k) = n!/(k!(n-k)!).A right cylinder has radius 7 and height 3. What is its total surface area? (in terms of π)
Correct answer: C
Surface area = 2πr2 + 2πrh = 2π(72) + 2π(7)(3) = 98π + 42π = 140π. Answer C. Key: cylinder surface = two bases + lateral area.Given x + y = 35 and xy = 306, what is x2 + y2?
Correct answer: D
x2 + y2 = (x+y)2 - 2xy = 352 - 2×306 = 1225 - 612 = 613. Answer D. Key: identity linking sum of squares to sum and product.In how many ways can 35 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 = 35. For a=1: 17 ways — (1,1,33) to (1,17,17); a=2: 15 ways — (2,2,31) to (2,16,17); a=3: 14 ways — (3,3,29) to (3,16,16); a=4: 12 ways — (4,4,27) to (4,15,16); a=5: 11 ways — (5,5,25) to (5,15,15); a=6: 9 ways — (6,6,23) to (6,14,15); a=7: 8 ways — (7,7,21) to (7,14,14); a=8: 6 ways — (8,8,19) to (8,13,14); a=9: 5 ways — (9,9,17) to (9,13,13); a=10: 3 ways — (10,10,15) to (10,12,13); a=11: 2 ways — (11,11,13) to (11,12,12). Total 102. Answer E. Key: ordered enumeration to avoid duplicates.Unified AMC 10 specs, consistent across years.
2000 has a single paper; AMC 10 added Papers A and B from 2002, when students may take either or both and keep the higher score.
Four steps · Overview / Topics / Practice / Prep
Read the 2000 AMC 10 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 2000 AMC 10 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 2000 AMC 10 papers and answers.
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