Difference between revisions of "1984 AIME Problems"
Line 30: | Line 30: | ||
== Problem 7 == | == Problem 7 == | ||
+ | The function f is defined on the set of integers and satisfies | ||
+ | <math> | ||
+ | f(n)= | ||
+ | \begin{cases} | ||
+ | n-3 & n\ge 1000 \\ | ||
+ | f(f(n+5)) & n<1000 | ||
+ | \end{cases} | ||
+ | </math> | ||
+ | Find <math>\displaystyle f(84)</math>. | ||
[[1984 AIME Problems/Problem 7|Solution]] | [[1984 AIME Problems/Problem 7|Solution]] |
Revision as of 23:56, 20 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 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 .