Difference between revisions of "1985 AIME Problems"
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==Problem 12== | ==Problem 12== | ||
+ | Let <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math> be the [[vertex | vertices]] of a regular [[tetrahedron]] each of whose [[edge]]s measures 1 meter. A bug, starting from vertex <math>A</math>, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let <math>p = \frac n{729}</math> be the [[probability]] that the bug is at vertex <math>A</math> when it has crawled exactly 7 meters. Find the value of <math>n</math>. | ||
+ | [[1985 AIME Problems/Problem 12 | Solution]] | ||
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==Problem 13== | ==Problem 13== | ||
Revision as of 02:25, 21 January 2007
Contents
Problem 1
Let , and for
let
. Calculate the product
.
Problem 2
When a right triangle is rotated about one leg, the volume of the cone produced is . When the triangle is rotated about the other leg, the volume of the cone produced is
. What is the length (in cm) of the hypotenuse of the triangle?
Problem 3
Find if
,
, and
are positive integers which satisfy
, where
.
Problem 4
A small square is constructed inside a square of area 1 by dividing each side of the unit square into equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of
if the the area of the small square is exactly
.
Problem 5
A sequence of integers is chosen so that
for each
. What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492?
Problem 6
As shown in the figure, triangle is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle
.
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
Problem 7
Assume that ,
,
, and
are positive integers such that
,
, and
. Determine
.
Problem 8
The sum of the following seven numbers is exactly 19: ,
,
,
,
,
,
. It is desired to replace each
by an integer approximation
,
, so that the sum of the
's is also 19 and so that
, the maximum of the "errors"
, the maximum absolute value of the difference, is as small as possible. For this minimum
, what is
?
Problem 9
In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of ,
, and
radians, respectively, where
. If
, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?
Problem 10
How many of the first 1000 positive integers can be expressed in the form
,
where is a real number, and
denotes the greatest integer less than or equal to
?
Problem 11
An ellipse has foci at and
in the
-plane and is tangent to the
-axis. What is the length of its major axis?
Problem 12
Let ,
,
and
be the vertices of a regular tetrahedron each of whose edges measures 1 meter. A bug, starting from vertex
, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let
be the probability that the bug is at vertex
when it has crawled exactly 7 meters. Find the value of
.