simula, 193 bytes

Result

AC

Code [DL]

begin integer a,b,r,i;while i<50do begin a:=inint;b:=0;r:=1;while a>0do begin b:=b+r*(a-a//10*10);r:=r*2;a:=a/10;end;a:=sqrt(b*8+1);if a*a=b*8+1then outchar('1')else outchar('0');i:=i+1;end;end

stdin

01011011
00010101
01000010
11100111
11111101
00011100
01100100
01101010
01111001
00000101
00000000
01101001
01101111
10100011
10111011
00000001
10110101
01011011
01001110
00100100
00101101
11010100
00100100
11001100
11010010
00001010
11011100
00001111
10011110
01101000
10011001
01011010
00000110
10100010
10101011
01111000
00000011
11100110
01011100
11110011
11011010
10111110
01000100
11101111
00110111
10001000
10110010
10001010
10111111
01011000

stdout

11111100001100010111101011010010101110000100110000

stderr

Compiling /tmp/code.sim:
gcc -g -O2 -c code.c
code.c: In function ‘main’:
code.c:115:136: warning: implicit declaration of function ‘exit’ [-Wimplicit-function-declaration]
 int main(int __argc, char *__argv[]){__progname= __argv[0];goto __sw;__l0:__rstart(__argc,__argv);goto __start;__l1:__slutt:__rslutt();exit(0);__start:__curent_map=__mapmain;
                                                                                                                                        ^
code.c:115:136: warning: incompatible implicit declaration of built-in function ‘exit’
code.c:115:136: note: include ‘<stdlib.h>’ or provide a declaration of ‘exit’
/tmp/code.sim:1:954: warning: implicit declaration of function ‘__rbe’ [-Wimplicit-function-declaration]
/tmp/code.sim: In function ‘__init’:
/tmp/code.sim:4:19: warning: implicit declaration of function ‘__init_FILE’ [-Wimplicit-function-declaration]
/tmp/code.sim: At top level:
/tmp/code.sim:5:1: warning: return type defaults to ‘int’ [-Wimplicit-int]
/tmp/code.sim:13:1: warning: return type defaults to ‘int’ [-Wimplicit-int]
gcc -g -O2 -o /tmp/code code.o -L/usr/local/lib -lcim

strace