1. use strict and use warnings.
2. I don't think that your trick of $line % 2 == 0 can work; both singles and pairs can occur on even or odd lines.
3. If you "pre-load" $prev before you begin your while(<>){...} loop, you can avoid keeping track of $line == 1 and $line >= 2.

Here is my (loosely tested) version:

#!perl
use strict;
use warnings;

my $last_line = <DATA>;
my $seen_count = 1;

while (<DATA>) {
    if ( $last_line ne $_ ) {
        print $last_line if $seen_count == 1;
        $last_line = $_;
        $seen_count = 0;
    }
    $seen_count++;
}
print $last_line if $seen_count == 1;

__END__
a1a
a1a
b1b
c1c
c1c
d1d
d1d
e1e
f1f
g1g
g1g
h1h
h1h
i1i
j1j
[download]