2000

2000 AMC12 Paper & Solutions Pick

25 questions with solutions, millennium exam, moderate difficulty.

Sample Problems Download PDF

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25 Questions

75 Minutes

Max Score 150

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Exam Overview

This exam emphasizes multiple math topics. Overall difficulty is moderate.

DDifficulty

EasyQ1-10
MediumQ11-20
HardQ21-25

TTopics

Algebra35%

Geometry30%

Number Theory15%

Combinatorics20%

AAwards

AIME Qualification

AIME Qualification (Top 2.5%)

85+

Honor Roll

Honor Roll (Top 5%)

75+

Achievement Roll

Grade 10 and below

60+

Sample Problems

2000 AMC12 Sample Problems (12 questions)

Sample reference problems by difficulty — click an option to check your answer

Q1EasyPolynomial

If f(x) = 2x² - 3x + 1, find f(2).

A) 3

B) 1

C) 5

D) 7

E) 9

Steps

1f(2) = 2×2² - 3×2 + 1

2= 2×4 - 6 + 1 = 3

Answer: ASubstitute, then exponentiate, multiply/divide, add/subtract

Q3EasyArithmetic Seq

Arithmetic sequence: a₁=3, d=2, find the 10-th term.

A) 15

B) 21

C) 18

D) 24

E) 27

Steps

1aₙ = a₁ + (n-1)d

2a_10 = 3 + 9×2 = 21

Answer: BGeneral term: aₙ=a₁+(n-1)d

Q5EasyExponent

If 2ˣ = 8, find x.

A) 1

B) 2

C) 3

D) 4

E) 5

Steps

18 = 2^3

2so x = 3

Answer: CConvert to same base, compare exponents

Q7EasyVieta's

Sum of roots of x² - 5x + 4 = 0?

A) 1

B) 3

C) 7

D) 5

E) 9

Steps

1Vieta's:sum of roots = -(-5)/1 = 5

2product of roots = 4

Answer: DFor x²-px+q=0, sum of roots = p

Q10MediumDivisibility

How many integers from 1 to 30 are divisible by 3?

A) 6

B) 8

C) 12

D) 14

E) 10

Steps

1⌊30 / 3⌋ = 10

Answer: ECount divisible by k: ⌊n/k⌋

Q12MediumDiagonals

How many diagonals does a regular 7-gon have?

A) 14

B) 8

C) 11

D) 17

E) 20

Steps

1diagonals = n(n-3)/2

2= 7×4/2 = 14

Answer: An-gon diagonals: n(n-3)/2

Q15MediumCombination

Compute C(6, 3).

A) 14

B) 20

C) 17

D) 23

E) 26

Steps

1C(n,3) = n(n-1)(n-2)/6

2= 6×5×4/6 = 20

Answer: BCombination: C(n,k)=n!/(k!(n-k)!)

Q17MediumPythagorean

Right triangle legs 3 and 4, find the hypotenuse.

A) 1

B) 3

C) 5

D) 7

E) 9

Steps

1c² = 3² + 4² = 9 + 16 = 25

2c = √25 = 5

Answer: CPythagorean theorem a²+b²=c²

Q20HardModular Arith

Find the remainder of 3^4 divided by 5.

A) 0

B) 2

C) 3

D) 1

E) 4

Steps

1compute 3^4 mod 5

2simplify step by step (modular arithmetic)

3=1

Answer: DModular arithmetic simplifies large exponents

Q22HardCompleting Sq

Find the minimum of f(x) = x² - 4x + 7.

A) 1

B) 5

C) 7

D) 9

E) 3

Steps

1complete the square:f(x) = (x - 2)² + 3

2when x = 2 , min at 3

Answer: ECompleting the square for quadratic extrema

Q24HardInclusion-Excl

4 distinct balls into 3 distinct boxes, each ≥1 ball, how many ways?

A) 36

B) 26

C) 31

D) 41

E) 46

Steps

1total 3^4 = 81

2subtract empty boxes: -C(3,1)×2^4 = -48

3add back 2 empty boxes: +C(3,2)×1 = +3

4total 81 - 48 + 3 = 36

Answer: AInclusion-exclusion handles 'at least'

Q25HardLaw of Cosines

In △ABC, a=5, b=6, cos C=1/2, find c².

A) 23

B) 31

C) 27

D) 35

E) 39

Steps

1Law of cosines c² = a² + b² - 2ab·cos C

2= 25 + 36 - 30 = 31

Answer: BLaw of cosines is key for solving triangles

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2000 AMC12 Sample Problems (12 questions)

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