Difference between revisions of "2013 Canadian MO Problems/Problem 4"
| Line 27: | Line 27: | ||
<math>g_j(1) =\min (\lceil j\rceil, n)+\min (\lceil j\rceil, n)=2\lceil j\rceil</math> | <math>g_j(1) =\min (\lceil j\rceil, n)+\min (\lceil j\rceil, n)=2\lceil j\rceil</math> | ||
| + | |||
| + | Since <math>j</math> is integer we have: | ||
<math>\sum_{j=1}^n g_j(1)=2\sum_{j=1}^n \lceil j\rceil =2\sum_{j=1}^n j = n^2+n</math> | <math>\sum_{j=1}^n g_j(1)=2\sum_{j=1}^n \lceil j\rceil =2\sum_{j=1}^n j = n^2+n</math> | ||
Revision as of 17:48, 27 November 2023
Problem
Let 
be a positive integer. For any positive integer 
and positive real number 
, define
![\[f_j(r) =\min (jr, n)+\min\left(\frac{j}{r}, n\right),\text{ and }g_j(r) =\min (\lceil jr\rceil, n)+\min\left(\left\lceil\frac{j}{r}\right\rceil, n\right),\]](http://latex.artofproblemsolving.com/5/9/8/59838efe579d7932f02643d88449c449d93f7d59.png)
where 
denotes the smallest integer greater than or equal to 
. Prove that
![\[\sum_{j=1}^n f_j(r)\leq n^2+n\leq \sum_{j=1}^n g_j(r)\]](http://latex.artofproblemsolving.com/6/9/3/69378627bf482631c8a7157b50157ae7a86d3b97.png)
for all positive real numbers .
Solution
First thing to note on both functions is the following:
and 
Thus, we are going to look at two cases:\. When 
, and when 
which is the same as when 
Case 1:
Since 
in the sum, then
and the equality holds.
Likewise,
Since is integer we have:
~Tomas Diaz. orders@tomasdiaz.com Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
