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My father, you see, was frankly quite gullible. Many years ago he fell for one of these "investment" scams. The scam promised to exactly double your investment every month, and excited by that, my father spent all his money starting such an account. And not only that. He immediately took out two loans which he used to start two more accounts.

To this very day, the scammers had sent him an email every month for each of the accounts, writing out the current balance in dollars and cents. My father in turn had printed all these emails, and these printouts were stacked on his desk.

Yesterday, I had looked at all the printouts for the past 2 years. As you can imagine, the numbers were now so ridiculously large that I didn't bother looking at the dollar amounts. But I noticed a curious property of the amount of cents: Despite the large number of printouts, the number of unique cent values wasn't nearly as large. In fact, there were only a modestly large square number of different values.

Finding this a bit curious, I wrote down the largest cent value written for each of the accounts, and the sum of these three values.

The next day, I had planned to go through the rest of stack. But unfortunately, my husband had thrown it away, junk as it admittedly was. He had spared the first printout for each account though, the one with the balance after the first supposed doubling had taken place.

When taking the cent amounts on those printouts into account as well, I noticed that the sum of the maximum cent amount for each account became 12 cent higher than yesterday.

What were the exact amount of cents on those three remaining printouts?

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    $\begingroup$ I don't quite get it. What were the first numbers you wrote down? You say "the largest cent value written for each of the accounts", but if that's the largest then it must be at least as large as whatever's written on the remaining printouts. Or did you consider only some specific subset of the printouts on the first day? $\endgroup$ – Rand al'Thor Jul 3 at 19:07
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    $\begingroup$ The first set consists of the printouts from the last two years, the second set consists of the very first printouts, which are much older. $\endgroup$ – AxiomaticSystem Jul 3 at 20:13
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The three printouts have cent values of

0, 90, and 98.

Why?

There are three cycles that cent values can fall into:
A cycle of length 1, containing 0.
A cycle of length 4, containing multiples of 20.
A cycle of length 20, containing multiples of 4.
The newest printouts cover 24 months and thus the entirety of any cycle, so to have a square number of values we have to have one account in each cycle. Since the accounts are many years old, the last 2 years only contain values in these cycles, so the highest values on the sheets are the maxima of the cycles: 0, 80, and 96.
The old printouts must contain even numbers, since a double has already occurred, but the 0 account cannot have 50 cents and the 96 account can only yield us 2 (from 98) so we must get 10 more from the 80 account by starting at 90.

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