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 6 of 6: Finish

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n=6,\qquad m=36

Detailed analysis

For $n>1$ , $6^{n-1}>|n-6|$ unless $n=6$ (and the small cases can be checked directly); hence $n=6$ . Substitution gives $m=36$ , with six prizes awarded each day.