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Art of Problem Solving

2005 AMC 8 Problems

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Contents

1 Problem 1

2 Problem 2

3 Problem 3

4 Problem 4

5 Problem 5

6 Problem 6

7 Problem 7

8 Problem 8

9 Problem 910 Problem 1011 Problem 1112 Problem 1213 Problem 1314 Problem 1415 Problem 1516 Problem 1617 Problem 1718 Problem 1819 Problem 1920 Problem 2021 Problem 21

22 Problem 2223 Problem

2324 Problem

2425 Problem 2526 See Also

Problem 1

Connie m ultiplies a num ber by 2 and gets 60 as her answer. However, she should have divided the num ber by 2 to get the correct answer. What is the correct answer?

Solution

Problem 2

Karl bought five folders from Pay-A-Lot at a cost of each. Pay-A-Lot had a 20%-off sale the following day. How m uch could Karl have saved on the purchase by waiting a day?

Solution

Problem 3

What is the m inim um num ber of sm all squares that m ust be colored black so that a line of sym m etry lies on

the diagonal of square ?

Solution

Problem 4

A square and a triangle have equal perim eters. The lengths of the three sides of the triangle are 6.1 cm,

8.2 cm and 9.7 cm. What is the area of the square in square centim eters?

Solution

Problem 5

Soda is sold in packs of 6, 12 and 24 cans. What is the m inim um num ber of packs needed to buy exactly 90 cans of soda?

Solution

Problem 6

Suppose is a digit. For how m any values of is ?

Solution

Problem 7

Bill walks m ile south, then m ile east, and finally m ile south. How m any m iles is he, in a direct line, from his starting point?

Solution

Problem 8

Suppose m and n are positive odd integers. Which of the following m ust also be an odd integer?

Solution

Problem 9

In quadrilateral , sides and both have length 10, sides and both have length 17,

and the m easure of angle is . What is the length of diagonal ?

Solution

Problem 10

Joe had walked half way from hom e to school when he realized he was late. He ran the rest of the way to school. He ran 3 tim es as fast as he walked. Joe took 6 m inutes to walk half way to school. How m any

m inutes did it take Joe to get from hom e to school?

Solution

Problem 11

The sales tax rate in Bergville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its $90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up $90.00 and adds 6% sales tax, then subtracts 20% from this total. Jill rings up $90.00, subtracts 20% of the price, then adds 6% of the discounted price for sales tax. What is Jack's total m inus Jill's total?

Solution

Problem 12

Big Al, the ape, ate 100 bananas from May 1 through May 5. Each day he ate six m ore bananas than on the previous day. How m any bananas did Big Al eat on May 5?

Solution

Problem 13

The area of polygon is 52 with , and . What is ?

Solution

Problem 14

The Little Twelve Basketball Conference has two divisions, with six team s in each division. Each team plays each of the other team s in its own division twice and every team in the other division once. How m any conference gam es are scheduled?

Solution

Problem 15

How m any different isosceles triangles have integer side lengths and perim eter 23?

Solution

Problem 16

A five-legged Martian has a drawer full of socks, each of which is red, white or blue, and there are at least five socks of each color. The Martian pulls out one sock at a tim e without looking. How m any socks m ust the Martian rem ove from the drawer to be certain there will be 5 socks of the sam e color?

The results of a cross-country team's training run are graphed below. Which student has the greatest average speed?

Solution

Problem 18

How m any three-digit num bers are divisible by 13?

Solution

Problem 19

What is the perim eter of trapezoid ?

Solution

Problem 20

Alice and Bob play a gam e involving a circle whose circum ference is divided by 12 equally-spaced points. The points are num bered clockwise, from 1 to 12. Both start on point 12. Alice m oves clockwise and Bob, counterclockwise. In a turn of the gam e, Alice m oves 5 points clockwise and Bob m oves 9 points counterclockwise. The gam e ends when they stop on the sam e point. How m any turns will this take?

Solution

Problem 21

How m any distinct triangles can be drawn using three of the dots below as vertices?

Solution

Problem 22

A com pany sells detergent in three different sized boxes: sm all (S), m edium (M) and large (L). The m edium size costs 50% m ore than the sm all size and contains 20% less detergent than the large size. The large size contains twice as m uch detergent as the sm all size and costs 30% m ore than the m edium size. Rank the three sizes from best to worst buy.

Isosceles right triangle encloses a sem icircle of area . The circle has its center on hypotenuse

and is tangent to sides and . What is the area of triangle ?

Solution

Problem 24

A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator autom atically displays the result. For instance, if the calculator originally displayed "9" and you pressed [+1], it would display "10." If you then pressed [x2], it would display "20." Starting with the display "1," what is the fewest num ber of keystrokes you would need to reach "200"?

Solution

Problem 25

A square with side length 2 and a circle share the sam e center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?

Solution