In a sports competition, there were M medals awarded on N successive days, where N is greater than 1.
- On the first day: 1 Medal and 1/9 of the remaining (M-1) medals were awarded.
- On the second day: 2 Medals and 1/9 of the now remaining medals were awarded, ...and, so on.
- On the Nth and the last day: the remaining N medals were awarded.
How many days did the contest last and how many medals were awarded altogether?
Provide valid reasoning for your answers.
I will generalize 1/9 into 1/K (K>=3) and show that M=(K-1)^2 and N=K-1 solves the problem where K-1 medals get awarded each day.
I will use induction to show K-1 medals are awarded each day.
The base case will be Day 1. On Day 1 all (K-1)^2 medals are available. Then 1 + [1/K]*[(K-1)^2-1] = 1 + (K^2-2K)/K = K-1 medals are awarded. This leaves (K-1)^2-(K-1) = (K-1)*(K-2) medals available for the Day 2.
Suppose on day D we have (K-1)*(K-D) medals available to award. Then D + [1/K]*[(K-1)*(K-D)-D] = D + [K^2-KD-K]/K = D + K-D-1 = K-1 medals are awarded. This leaves (K-1)*(K-D)-(K-1) = (K-1)*(K-(D+1)) medals available for the next day.
(K-1)*(K-(D+1)) is the needed number of medals for Day D+1, and Day 1 leaves the appropriate number of medals for the induction to apply, so we can conclude the induction holds true and K-1 medals are awarded for each day.
Last thing to check is the final day. On Day K-1 there are (K-1)*(K-(K-1)) = K-1 medals remaining to award. Since awarding K-1 medals this day awards all medals we conclude that the contest lasted K-1 days. This also checks with the starting (K-1)^2 medals and dividing by K-1 medals per day.
If we are to plug this result back into K=9 we get 64 medals awarded over 8 days.
Solution
