Mock AIME 1 Pre 2005 Problems/Problem 3
Problem
and
are collinear in that order such that
and
. If
can be any point in space, what is the smallest possible value of
?
Solution
Let the altitude from onto
at
have lengths
and
. It is clear that, for a given
value,
,
,
,
, and
are all minimized when
. So
is on
, and therefore,
. Thus,
=r,
,
,
, and
Squaring each of these gives:
This reaches its minimum at , at which point the sum of the squares of the distances is
.
See also
Mock AIME 1 Pre 2005 (Problems, Source) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |