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You're on a mathematical Olympiad, there are

**m**medals and it lasts for

**n**days.

First day committee gives [itex]U_{1}=1+\frac{1}{7}(m-1)[/itex] medals.

On the second day [itex]U_{2}=2+\frac{1}{7}(m-2-U_{1})[/itex] medals, and so on...

On the last day [itex]U_{n}=n[/itex].

Attempt at the solution:

My assumption is: [tex]U_{i}=i+\frac{1}{7}(m-i-\sum_{j=1}^{i}U_{j}+U_{i})[/tex]

If i take the difference of [itex]U_{i}-U_{(i-1)}[/itex],

I find this way to write the recursion:[tex]U_{i}=\frac{6}{7}(1+U_{(i-1)}),\;(i=2,3,...,n)[/tex]

If I set [itex]i=n[/itex] and do some algebra, then I get the following:[tex]13n=6+6m\;\; (1)[/tex]

...and this is where I'm not sure what to do, number of medals must be discrete...

Every [itex]U_{i}[/itex] must be discrete.

I tried from equation (1) to derive what

**m**and

**n**must be:[tex]n=6k \\ m=-1+13k,\;k\epsilon \mathbb{N}[/tex] However I am not sure how to capture this notion that every [itex]U_{i}[/itex] must be discrete.

Thank you for reading. :)