Deal with the problem a little at a time. First bit is loop over the elements (with strictures, declarations and initialisation thrown in):

use warnings;
use strict;

my @times = (5, 10, 20, 25, 30, 40, 45, 50, 55, 65, 75, 80, 90);
my @times1;
my @times2;

for my $elmt (@times) {
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Then deal with the first case (and the special case for the first element):

    if (! @times1 || $elmt - $times1[-1] >= 10) {
        push @times1, $elmt;
        next;
    }
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Then deal with the second case (and again the special case for the first special element):

    if (! @times2 || $elmt - $times2[-1] >= 10) {
        push @times2, $elmt;
        next;
    }
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Then deal with the exception (and close the loop body):

    push @times1, $elmt;
}
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Finally print the result:

print "Times1: @times1\n";
print "Times2: @times2\n";
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Prints:

Times1: 5 20 30 40 50 65 75 90
Times2: 10 25 45 55 80
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