1996 AIME Problems
Contents
Problem 1
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?
Problem 3
Find the smallest positive integer for which the expansion of
, after like terms have been collected, has at least 1996 terms.
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
.
Problem 5
Suppose that the roots of are
,
, and
, and that the roots of
are
,
, and
. Find
.
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
.
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?
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
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?