Problem 6

In a sports contest,

m

medals were awarded on

n

successive days (

n>1

). On the first day, one medal and

\frac17

of the remaining

m-1

medals were awarded. On the second day, two medals and

\frac17

of the then remaining medals were awarded, and so on. On the

n

-th and last day, the remaining

n

medals were awarded. Find

m

and

n

Step 1 of 6: Track the remainder

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m_r=\text{prizes remaining at the start of day }r,\quad m_1=m

Detailed analysis

Let $m_r$ be the number of prizes at the start of day $r$ , with $m_1=m$ the total. On the final day, the rule for day $n$ and the fact that all remaining prizes are awarded give $m_n=n$ .