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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>.
  
 
[[1996 AIME Problems/Problem 6|Solution]]
 
[[1996 AIME Problems/Problem 6|Solution]]

Revision as of 14:50, 24 September 2007

Problem 1

Solution

Problem 2

For each real number $x$, let $\lfloor x \rfloor$ denote the greatest integer that does not exceed x. For how man positive integers $n$ is it true that $n<1000$ and that $\lfloor \log_{2} n \rfloor$ is a positive even integer?

Solution

Problem 3

Find the smallest positive integer $n$ for which the expansion of $(xy-3x+7y-21)^n$, 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 $x$ 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 $1000x$.

Solution

Problem 5

Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a$, $b$, and $c$, and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$, $b+c$, and $c+a$. Find $t$.

Solution

Problem 6

In a five-team tournament, each team plays one game with every other team. Each team has a $50%$ (Error compiling LaTeX. Unknown error_msg) chance of winning any game it plays. (There are no ties.) Let $\dfrac{m}{n}$ be the probability that the tournament will product neither an undefeated team nor a winless team, where $m$ and $n$ are relatively prime integers. Find $m+n$.

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

A $150\times 324\times 375$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\times 1\times 1$ cubes?

Solution

Problem 15

Solution

See also

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