Low-tech solution which just uses a variable to indicate if the next line should be printed:

use warnings;
use strict;

my $print_next;
while (<DATA>) {
    print if $print_next;
    $print_next = /XXXX/;
}

__DATA__
ZZZZZZZZZZZZ
YYYYY
XXXX
XX
XXXX
3333
ii
XXXX
asdfddd
[download]