Difference between revisions of "2006 AIME A Problems/Problem 13"
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Revision as of 18:50, 25 September 2007
Problem
For each even positive integer , let
denote the greatest power of 2 that divides
For example,
and
For each positive integer
let
Find the greatest integer
less than 1000 such that
is a perfect square.
Solution
Given , consider
. Define
. There are
elements of
that are divisible by
,
elements of
that are divisible by
but not by
and
elements of
that are divisible by
but not by
.
Thus , so we need
for $k \in \N$ (Error compiling LaTeX. Unknown error_msg). Now notice we also require
, so if
also (but
), then
, so we have
. If
, then
, so we have
. Finally,
could possibly be
or 256. The maximum possible
is thus
.
See also
2006 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |