simula, 185 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);r:=0;if a*a=b*8+1then r:=1;outint(r,0);i:=i+1;end;end

stdin

01011101
10111111
00010101
10111011
11000100
00101101
11010000
10101011
10001000
11110101
01000010
10111110
00100100
01101010
01111000
00000001
01001110
01100100
01101100
10101000
11110011
11111100
11100111
01101001
01111000
11110001
10011001
10110110
00110111
10001001
11010010
00010001
00001010
00001111
00011100
10111000
11000111
00000000
00000011
10110011
10101100
10001110
01000100
11100100
10101010
01011011
11111101
00101101
10010111
00000110

stdout

00100101101110111000001110101010111001100000011101

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:1:952: 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