Difference between revisions of "2013 Canadian MO Problems/Problem 4"
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==Solution== | ==Solution== | ||
| − | + | First thing to note on both functions is the following: | |
| − | + | <math>f_j(1/r) =\min (\frac{j}{r}, n)+\min\left(jr, n\right)=f_j(1/r)</math>, | |
| + | and g_j(1/r) =\min (\lceil frac{j}{r}\rceil, n)+\min\left(\left\lceil\jr\right\rceil, n\right) = f_j(r)<math> | ||
| + | |||
| + | |||
| + | Case 1: </math>r=1<math> | ||
| + | |||
| + | Since </math>j \le n$ in the sum, the | ||
| + | f_j(r) =\min (jr, n)+\min\left(\frac{j}{r}, n\right) | ||
~Tomas Diaz. orders@tomasdiaz.com | ~Tomas Diaz. orders@tomasdiaz.com | ||
{{olution}} | {{olution}} | ||
Revision as of 16:32, 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 g_j(1/r) =\min (\lceil frac{j}{r}\rceil, n)+\min\left(\left\lceil\jr\right\rceil, n\right) = f_j(r)
r=1
j \le n$ in the sum, the
f_j(r) =\min (jr, n)+\min\left(\frac{j}{r}, n\right)
~Tomas Diaz. orders@tomasdiaz.com Template:Olution
