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Asked: 2026-05-18T05:06:58+00:00

Here is my problem: There are n companies distributing products. All products should be

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Here is my problem:

  • There are n companies distributing
    products.
  • All products should be distributed in k days
  • Distributing products of company Ci should be consecutive – it means that it can be distributed on days 2,3,4,5 but not 2,3,6,7
  • number of distributed products by company Ci on day j should be less than (or equal) on day j-1 (if there were any on day j-1)
  • difference between distributed products between days i and j should not be greater than 1

Example:

We have 3 days to distribute products. Products of company A: a,a,a,a,a. Products of company B: b,b,b. Products of company C: c,c

Fair distribution:
[aab,aabc,abc]

Invalid distribution:
[aabc,aabc,ab]
because on 1st day there are 4 products, on 3rd day 2 products (difference > 1)

Invalid distribution:
[abc,aabc,aab]
because on 1st day there is one product A, and on 2nd day there are 2 products A, so distribution of product A is not non-decreasing

EDIT
if there is a case that makes fair distribution impossible please provide it with short description, I’ll accept the answer

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1 Answer

  1. Editorial Team

    Gareth Rees’s comment on djna’s answer is right — the following counterexample is unsolvable:

    • 3 days, 7 items from company A and 5 items from company B

    I tested this with the following dumbest-possible brute-force Perl program (which takes well under a second, despite being very inefficient):

    my ($na, $nb) = (7, 5);
for (my $a1 = 0; $a1 <= $na; ++$a1) {
    for (my $a2 = 0; $a2 <= $na - $a1; ++$a2) {
        my $a3 = $na - $a1 - $a2;
        for (my $b1 = 0; $b1 <= $nb; ++$b1) {
            for (my $b2 = 0; $b2 <= $nb - $b1; ++$b2) {
                my $b3 = $nb - $b1 - $b2;
                if ($a1 >= $a2 && $a2 >= $a3 || $a1 == 0 && $a2 >= $a3 || $a1 == 0 && $a2 == 0) {
                    if ($b1 >= $b2 && $b2 >= $b3 || $b1 == 0 && $b2 >= $b3 || $b1 == 0 && $b2 == 0) {
                        if (max($a1 + $b1, $a2 + $b2, $a3 + $b3) - min($a1 + $b1, $a2 + $b2, $a3 + $b3) <= 1) {
                            print "Success! ($a1,$a2,$a3), ($b1,$b2,$b3)\n";
                        }
                    }
                }
            }
        }
    }
}

    Please have a look and verify that I haven’t made any stupid mistakes. (I’ve omitted max() and min() for brevity — they just do what you’d expect.)

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