1997 AIME Problems

Problem 1

How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?

Solution

Problem 2

The nine horizontal and nine vertical lines on an checkeboard form rectangles, of which are squares. The number can be written in the form where and are relatively prime positive integers. Find

Solution

Problem 3

Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number?

Solution

Problem 4

Circles of radii 5, 5, 8, and are mutually externally tangent, where and are relatively prime positive integers. Find

Solution

Problem 5

The number can be expressed as a four-place decimal where and represent digits, any of which could be zero. It is desired to approximate by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to is What is the number of possible values for ?

Solution

Problem 6

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

See also

• American Invitational Mathematics Examination
• AIME Problems and Solutions
• Mathematics competition resources