my $i = 0;
while (<FILE>) {
  next if ++$i % 2;
  print;
}
[download]

while (<FILE>) {
  next if $. % 2;
  print;
}
[download]

my $toggle = 0;
while (<FILE>) {
   next if $toggle ^= 1;
   print;
}
[download]

By the way,
 #!usr/bin/perl
is wrong. You must remove the leading space, and you must make the path absolute (or else it will only work form the root):
#!/usr/bin/perl

Update: I didn't read your post close enough. The above didn't answer your question. A solution would be:

for (;;) {
   last if not defined (my $line1 = <FILE>);
   last if not defined (my $line2 = <FILE>);

   next if not $line1 =~ /red line/;

   print($line1, $line2);
}
[download]