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

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

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

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

   print($line1, $line2);
}