OK, so combining this together with no_slogan's advise that the magic while(<>) consumes @ARGV, you need to store @ARGV before the first loop and restore it after the first loop is done.
my @old_ARGV = @ARGV;
my $count;
$count++ while <>;
print "COUNT = $count\n";
@ARGV = @old_ARGV;
print while <>;
And as you see, it's a bit pointless to first chomp off the newline if you add it later again in the print - so I removed that.
You could make good use of local here as well, but I leave that as an exercise for the reader ... :)
-- Hofmator |