1996 AIME Problems

Problem 1

Solution

Problem 2

For each real number , let denote the greatest integer that does not exceed x. For how man positive integers is it true that and that is a positive even integer?

Solution

Problem 3

Find the smallest positive integer for which the expansion of , after like terms have been collected, has at least 1996 terms.

Solution

Problem 4

A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of a shadow, which does not include the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed .

Solution

Problem 5

Suppose that the roots of are , , and , and that the roots of are , , and . Find .

Solution

Problem 6

In a five-team tournament, each team plays one game with every other team. Each team has a chance of winning any game it plays. (There are no ties.) Let be the probability that the tournament will product neither an undefeated team nor a winless team, where and are relatively prime integers. Find .

Solution

Problem 7

Two squares of a checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane board. How many inequivalent color schemes are possible?

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

A rectangular solid is made by gluing together cubes. An internal diagonal of this solid passes through the interiors of how many of the cubes?

Solution

Problem 15

Solution

See also

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