Question 49
There are 2013 cards numbered $0$, $1$, $2$, $\ldots$, $2012$. Initially, all the cards are placed with the face with a written number down. Then, we perform for each $i = 1, 2, \dots, 2013$ the following operation $i$ starting with $i = 1$ and with increasing order ending up with $i = 2013$:
Operation $i$: Flip each of the $i$ cards having the number $\left\lfloor \frac{2013j}{i} \right\rfloor$ for $j = 0, 1, \dots, i - 1$.
How many cards are with their faces up (showing their numbers) at the end of all the operations?
Here we denote by $[r]$ for each real number $r$ the greatest integer less than or equal to $r$.
Operation $i$: Flip each of the $i$ cards having the number $\left\lfloor \frac{2013j}{i} \right\rfloor$ for $j = 0, 1, \dots, i - 1$.
How many cards are with their faces up (showing their numbers) at the end of all the operations?
Here we denote by $[r]$ for each real number $r$ the greatest integer less than or equal to $r$.