Here's a go without using a big regex. It looks for begin{} and end{} keys and stores the text between the two in an anon array in a hash indexed by that key.

use strict;
use warnings;

#-- hash that will be keyed by Latex element
my %in = ();

while ( <DATA> )
{
 if ( /\\begin\{(\S+?)\}/ )
 {
  #-- tell that we're in a block
  $in{$1}->{Status} = "in";

  #-- add a new element to the anon array containing this info
  push @{$in{$1}->{Text}}, "";
 }
 if ( /\\end\{(\S+?)\}/ )
 {
  $in{$1}->{Status} = "pending";
 }

 #-- now loop on all keys, see we are in that element.
 foreach my $key ( keys %in )
 {
  my $status = $in{$key}->{Status};
  if ( $status eq "in" || $status eq "pending" )
  {
   #-- add text to last element of the array
   $in{$key}->{Text}->[$#{$in{$key}->{Text}}] .= $_;
  }
  $in{$key}->{Status} = "out" if $status eq "pending";
 }
}

#-- write it out. Here loop over all possible keys. Could be restricte
+d to
#    'exertext', 'answers', 'soln'
foreach my $key ( keys %in )
{
 my $file = $key . ".tex";
 print "Creating file '$file'\n";
 open( FILE, '>', $file) or die "Cannot open file '$file' for writing:
+ $!\n";

 #-- write each element of array
 my $i = 1;
 foreach my $text ( @{$in{$key}->{Text}} )
 {
  print FILE "%% Starting Element $i\n";
  print FILE $text;
  $i++;
 }
 close FILE;
}
[download]

The output is

• instructions.tex
• soln.tex
• answer.tex
• exer.tex
• exertext.tex

With nested tags, the sub tags are included in the given tag (so that exer.tex includes exertext.tex, soln.tex, answer.tex). If you ran it on a full doc, presumably "document.tex" would be the same as your input doc.

- j