php, 68 bytes

Result

AC

Code [DL]

<?php for(;$c=fgets(STDIN);)echo +!(fmod(sqrt(8*bindec($c)+1),1)>0);

stdin

00101101
00110111
10001000
00000110
00001111
10001000
11010010
00111111
01000010
01101110
00001101
01011011
10100100
00011111
10011111
11110011
00001000
10111011
11100111
00011100
01101001
11110010
00100100
11010110
11000011
10011001
01010101
00001001
11101101
01010001
00001010
10101011
10101001
10011110
00110010
10100101
11111101
01011000
01001001
01001110
00101100
00000110
01000110
01111000
10111110
00000011
00011110
00000000
00000001
00010101

stdout

11111110100100000011101001000011000010010101110111

stderr

strace