2003 Indonesia MO Problems/Problem 3
Revision as of 23:55, 10 August 2018 by Rockmanex3 (talk | contribs) (Solution to Problem 3 -- floors and ceilings)
Problem
Find all real solutions of the equation .
[Note: For any real number ,
is the largest integer less than or equal to
, and
denote the smallest integer more than or equal to
.]
Solution
There are two cases to consider -- one where is positive and one where
is negative.
For the positive case, if then the equation results in
Since the equation does not have an integral solution,
If we let
That means
and solving the equation yields
For the negative case, if then the equation results in
This also does not have an integral solution, so
If we let
That means
and this equation also yields
In interval notation, the solutions of the equation are
See Also
2003 Indonesia MO (Problems) | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 4 |
All Indonesia MO Problems and Solutions |