2005 AIME II Problems/Problem 5
Problem
Determine the number of ordered pairs of integers such that
and
Solution
Note that the equation can be rewritten as: or
Multiplying through by
and factoring yields
. Therefore,
or
. Therefore, either
or
. For the case
, note that
and
. Thus, all values of
from 2 to 44 will work. For the case
, note that
while
. Therefore, for this case, all values of
from 2 to 12 work. There are
possibilities for the square case and
possibilities for the cube case. Thus, the answer is
.
See also
2005 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |