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Back then in 2013 when I was participating in International Olympiad in Informatics (IOI), I was stumbled on the puzzles on their newsletters. Even until now, I'm pretty clueless for these.

Here are two of them. Could you help me to crack it?

puzzle on newsletter #1

puzzle on newsletter #2

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    $\begingroup$ You may have better luck on the Mathematics page if you find that us puzzlers can't solve it :) $\endgroup$
    – elarr
    Aug 16, 2018 at 12:33
  • $\begingroup$ Adding to what @elarr commented above, I strongly recommend you post this on the Mathematics Stack Exchange (Math SE). $\endgroup$
    – Mr Pie
    Aug 16, 2018 at 12:36
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    $\begingroup$ @elarr Don't write us off just yet, we might be on to something :D $\endgroup$
    – Eutherpy
    Aug 16, 2018 at 13:43

3 Answers 3

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I think this can be solved without math knowledge:

("Empodia", 2004) is

the name of the 4th task of the 2004 IOI

so the solution is probably

the 4th task of some other IOI?

Edit:

I think the second solution could be

("Buses", 2008) - the "Buses" task of the 2008 IOI, since the task description matches the mathematical notation... I think. Here is the link to that year's tasks: IOI2008

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  • $\begingroup$ I was just about to comment this! (+1) :D I wasn't sure though, because this puzzle is only related to Empodia by name; the actual task for Empodia that year didn't appear to have anything to do with the mathematical notation that is associated to it here... $\endgroup$
    – El-Guest
    Aug 16, 2018 at 12:44
  • $\begingroup$ @El-Guest Yeah, I just realized that too :D I'm guessing the notation is equivalent to the written task, in which case we do need mathematical knowledge (big time) to solve this... $\endgroup$
    – Eutherpy
    Aug 16, 2018 at 12:47
  • $\begingroup$ Eutherpy, I think you're more correct than I am...the more I look at Empodia, the more that I'm starting to see similarities between that task and this notation. $\endgroup$
    – El-Guest
    Aug 16, 2018 at 12:52
  • $\begingroup$ Aah, I see now.. So all of those mathematical notations describe past IOI problems (and we need to find it), and indeed the task description - in plain language - is preferred to be read rather than the notations! Thanks a lot!! :D. Btw I'm accepting this answer as it's the first that hinted the way to solve the puzzle. Also +1 for @ffao's answer for the second puzzle. $\endgroup$
    – athin
    Aug 17, 2018 at 0:40
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The second answer is

("Pairs", 2007)

Because the math states:

Given the dimensionality of the board $a = B$, a max distance $b = D$, and a set of animals $C$, how many pairs of animals can hear each other (that is, what is the size of the set of pairs of animals such that their distance is at most $b $)?

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  • $\begingroup$ I think I sort of get what the first one says, but I don't know enough IOI problems to know what they framed it as... $\endgroup$
    – ffao
    Aug 16, 2018 at 19:57
  • $\begingroup$ The one from Croatia, yes! I stumbled upon that one and it caught my eye, but I wasn't sure and decided not to answer :) $\endgroup$
    – Eutherpy
    Aug 16, 2018 at 21:35
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To add my two cents to @Eutherpy's answer, in plain language it appears that

given an integer $k$ and an invertible (apparently on $\mathbb{Z}$) function that maps the large interval $[1,k]$ to itself, we're trying to find the biggest set of intervals where (a) no element appears in more than one interval in the set, and (b) when you apply the function $\pi$ to any element in any interval, the interval remains the same size but shifts up.

An example for k = 5 (not the correct solution) seems to be:

using the function $\pi(x) = x+1$, where if we assume $a<b$ as is common for intervals, we could have the set $\{ [1,2] , [3,4] \}$ be maximal. Facetiously, since there is no $a<b$ restrictiion, the maximal solution would be $\{[1,1] , [2,2], [3,3], [4,4] \}$...but I have no idea on how that relates to the framed intervals from the Empodia problem.

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