1976 IMO Problems/Problem 3
Problem
A box whose shape is a parallelepiped can be completely filled with cubes of side If we put in it the maximum possible number of cubes, each of volume
, with the sides parallel to those of the box, then exactly
percent from the volume of the box is occupied. Determine the possible dimensions of the box.
Solution
The first statement tells us that the box has integer dimensions. Let the dimensions be ,
, and
, where they're each greater than 1. The second statement tells us that if
boxes with side length
are put into the box, that takes up
units of area. Thus
. Now the range of values that the LHS can take up is
to
, exclusive, since the fractions are not integers. Therefore we must find all
,
, and
such that
The is redundant, so we can eliminate that. We simplify the LHS:
See also
1976 IMO (Problems) • Resources | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
All IMO Problems and Solutions |