Difference between revisions of "1984 AIME Problems"
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== Problem 9 == | == Problem 9 == | ||
+ | In tetrahedron <math>\displaystyle ABCD</math>, edge <math>\displaystyle ABC</math> has length 3 cm. The area of face <math>\displaystyle AMC</math> is <math>\displaystyle 15\mbox{cm}^2</math> and the area of face <math>\displaystyle ABD</math> is <math>\displaystyle 12 \mbox { cm}^2</math>. These two faces meet each other at a <math>30^\circ</math> angle. Find the volume of the tetrahedron in <math>\displaystyle \mbox{cm}^3</math>. | ||
[[1984 AIME Problems/Problem 9|Solution]] | [[1984 AIME Problems/Problem 9|Solution]] |
Revision as of 00:16, 21 January 2007
Contents
[hide]Problem 1
Find the value of if , , is an arithmetic progression with common difference 1, and .
Problem 2
The integer is the smallest positive multiple of such that every digit of is either or . Compute .
Problem 3
A point is chosen in the interior of such that when lines are drawn through parallel to the sides of , the resulting smaller triangles , , and in the figure, have areas , , and , respectively. Find the area of .
Problem 4
Let be a list of positive integers - not necessarily distinct - in which the number appears. The arithmetic mean of the numbers in is . However, if is removed, the arithmetic mean of the numbers is . What's the largest number that can appear in ?
Problem 5
Determine the value of if and .
Problem 6
Three circles, each of radius 3, are drawn with centers at , , and . A line passing through is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?
Problem 7
The function f is defined on the set of integers and satisfies
Find .
Problem 8
The equation has complex roots with argument between and in thet complex plane. Determine the degree measure of .
Problem 9
In tetrahedron , edge has length 3 cm. The area of face is and the area of face is . These two faces meet each other at a angle. Find the volume of the tetrahedron in .