Niloy's Money Machine


Niloy gives you a money-making machine. If you give this machine some taka, it will return you some taka. If you give the machine a certain amount of taka and 7 taka more, it will return you the double of that certain amount of taka. What amount of taka can you get the most from this machine if you have $2023$ taka?


Proof Based Problems  


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Solution

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We can get an infinite amount of Taka from the machine.


Let $n_k$ denote the amount of money we have after $k$ exchanges with the machine. If we give the machine $x_k$ Taka s.t. $x_k>7$ in the $k$'th exchange, then the machine will return $2(x_k-7)=2x_k-14$ Taka. So, \[n_k=n_{k-1}-x_k+2x_k-14=n_{k-1}+x_k-14\]


So, if $x_k\geq 15$, then $n_k\geq n_{k-1}+1$.

We have $2023$ Taka in the beginning, so we can clearly give the machine 15 Taka in every move. So the total amount of money always increases by at least $1$.


$\therefore$ We can get an infinite amount of money.

This is a proof based problem added for learning purposes and does not accept submissions.

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