Difference between revisions of "1996 AIME Problems"
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== Problem 6 == | == Problem 6 == | ||
− | In a five-team tournament, each team plays one game with every other team. Each team has a <math>50%</math> chance of winning any game it plays. (There are no ties.) Let <math>\dfrac{m}{n}</math> be the probability that the tournament will product neither an undefeated team nor a winless team, where <math>m</math> and <math>n</math> are relatively prime integers. Find <math>m+n</math>. | + | In a five-team tournament, each team plays one game with every other team. Each team has a <math>50\%</math> chance of winning any game it plays. (There are no ties.) Let <math>\dfrac{m}{n}</math> be the probability that the tournament will product neither an undefeated team nor a winless team, where <math>m</math> and <math>n</math> are relatively prime integers. Find <math>m+n</math>. |
[[1996 AIME Problems/Problem 6|Solution]] | [[1996 AIME Problems/Problem 6|Solution]] |
Revision as of 15:50, 24 September 2007
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
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?