Difference between revisions of "1984 AIME Problems"
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== Problem 14 == | == Problem 14 == | ||
+ | What is the largest even integer that cannot be written as the sum of two odd composite numbers? | ||
[[1984 AIME Problems/Problem 14|Solution]] | [[1984 AIME Problems/Problem 14|Solution]] |
Revision as of 01:41, 21 January 2007
Contents
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 , 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
.
Problem 10
Mary told John her score on the American High School Mathematics Examination (AHSME), which was over . From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over
, John could not have determined this. What was Mary's score? (Recall that the AHSME consists of
multiple choice problems and that one's score,
, is computed by the formula
, where
is the number of correct answers and
is the number of wrong answers. Students are not penalized for problems left unanswered.)
Problem 11
A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let in lowest terms be the probability that no two birch trees are next to one another. Find
.
Problem 12
A function is defined for all real numbers and satisfies
and
for all
. If
is a root for
, what is the least number of roots
must have in the interval
?
Problem 13
Find the value of
Problem 14
What is the largest even integer that cannot be written as the sum of two odd composite numbers?