php, 66 bytes

Result

WA

Code [DL]

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

stdin

10001000
00000000
00010101
11111111
01000010
00111100
00100100
01011011
10111110
11100111
00011010
10101000
01101000
00001111
00100111
01111010
10000001
10100010
10010000
00011100
11001001
11100001
01000101
00000001
10010110
10101011
00111010
10111111
01001111
00110111
01111000
00101101
01101100
01101001
11011011
10001111
01001110
10100000
11010010
01010010
11000111
00000110
00000011
00000011
10011001
10011110
10001011
00001010
01001110
11111101

stdout

00010100001110111110111010111000101101011000011000

stderr

strace