2005

2005 AMC8 Paper & Solutions Pick

25 questions with solutions, moderate difficulty, balanced topic distribution.

Sample Problems Download PDF

Scan to get free 2005 PDF + solutions

12 Sample Questions

40 Minutes

Max Score 25

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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

Distinguished Honor Roll

Distinguished Honor Roll (Top 1%)

19+

Honor Roll

Honor Roll (Top 5%)

15+

Achievement Roll

Grade 6 and below · 15+ points

15+

Sample Problems

2005 AMC8 Sample Problems (12 questions)

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

Q1EasyArithmetic

Calculate the value of 25×4.

A) 100

B) 75

C) 80

D) 90

E) 125

Steps

25×4=100

Answer: ABasic multiplication

Q4EasyKey concept: ratios

A:B=3:4 and B:C=2:5. What is A:B:C?

A) 3:4:10

B) 3:2:5

C) 6:8:20

D) 3:4:5

E) 6:4:5

Steps

Make the common middle term B match: A:B=3:4, B:C=2:5=4:10

So A:B:C = 3:4:10

Answer: AContinued ratio: make the shared middle term consistent

Q6EasyArea

What is the area of a square with side length 11?

A) 121

B) 110

C) 111

D) 119

E) 120

Steps

11²=121

Answer: APerfect square

Q9EasyPrime Numbers

How many prime numbers are there below 20?

A) 6

B) 7

C) 8

D) 9

E) 10

Steps

2, 3, 5, 7, 11, 13, 17, 19 — that's 8 of them

Answer: C1 is not a prime number

Q11MediumKey concept: equations

Three consecutive integers sum to 72. What is the largest one?

A) 25

B) 23

C) 24

D) 26

E) 27

Steps

3x=72 → x=24

Largest = 25

Answer: AThe middle number equals the average

Q14MediumArea

What is the area of a triangle with sides 3, 4, and 5?

A) 6

B) 5

C) 7

D) 8

E) 10

Steps

3-4-5 is a right triangle

Area = 1/2×3×4 = 6

Answer: A3-4-5 Pythagorean triple

Q16MediumProbability

A bag contains 4 red, 3 blue, and 2 green balls. If one ball is drawn, what is the probability it is not red?

A) 2/9

B) 5/9

C) 4/9

D) 6/9

E) 7/9

Steps

Not red = 3+2 = 5

P = 5/9

Answer: BComplementary probability

Q19MediumNumber Theory

Find the value of GCD(48, 72).

A) 12

B) 24

C) 16

D) 18

E) 36

Steps

48 = 2⁴×3, 72 = 2³×3²

GCD = 2³×3 = 24

Answer: BTake the lowest power of each common prime factor

Q21HardLogic

5 players compete in a tournament, and each player wins a different number of games. What is the most games anyone wins?

A) 2

B) 3

C) 5

D) 6

E) 4

Steps

Total games = C(5,2) = 10

The 5 players' win counts are all different: 0, 1, 2, 3, 4

Sum = 10 ✓

The most is 4 games

Answer: EThe combination C(n,2) gives the total number of games

Q22HardArea

A regular hexagon has area 54√3. What is its side length?

A) 4

B) 5

C) 7

D) 6

E) 8

Steps

Area = (3√3/2)a² = 54√3

a² = 36 → a = 6

Answer: DArea formula for a regular hexagon

Q24HardKey concept: permutations

5 married couples (10 people) stand in a row, with each couple standing next to each other. How many arrangements are there?

A) 7680

B) 15360

C) 19200

D) 3840

E) 38400

Steps

Treat each of the 5 couples as a block: 5! arrangements

Within each couple, 2 orders

5!×2⁵ = 120×32 = 3840

Answer: DBlock method + internal arrangements

Q25HardCombinatorics

From {1, 2, ..., 10}, choose 3 numbers that form an arithmetic sequence. How many ways are there?

A) 16

B) 18

C) 20

D) 22

E) 24

Steps

Common difference 1: (1,2,3)...(8,9,10) = 8 ways

Common difference 2: (1,3,5)...(6,8,10) = 6 ways

Common difference 3: 4 ways; common difference 4: 2 ways

8+6+4+2 = 20

Answer: CEnumerate by common difference

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2005 AMC8 Paper & Solutions Pick

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Exam Overview

DDifficulty

TTopics

AAwards

2005 AMC8 Sample Problems (12 questions)

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