Difference between revisions of "1996 AIME Problems"
(→Problem 6) |
(→Problem 6) |
||
| Line 24: | Line 24: | ||
== 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 14:50, 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?
