c-gcc, 92 bytes

Result

AC

Code [DL]

s[9];main(j,a,t){for(;gets(s);putchar(a+48))for(j=t=strtol(s,a=0,2),j++;j--;)a+=j*j+j==t*2;}

stdin

00111000
00001011
11010010
10111110
00000110
11000100
00000001
01000110
01000010
10011110
00011000
11010111
00110101
11111101
01100100
00100100
00001010
01010110
11011011
11100001
10101011
01101001
01010001
00000011
00011100
01110110
00000000
01011011
10000101
01110001
00101101
01001110
01010100
11100111
10011001
10011000
01010011
11001101
00001010
01111000
00001111
10001000
10110101
11011010
00010101
10100000
01100101
01111101
00110111
11100111

stdout

00111010100001011000110110110011011000111100100011

stderr

/tmp/code.c:1:1: warning: data definition has no type or storage class
 s[9];main(j,a,t){for(;gets(s);putchar(a+48))for(j=t=strtol(s,a=0,2),j++;j--;)a+=j*j+j==t*2;}
 ^
/tmp/code.c:1:1: warning: type defaults to ‘int’ in declaration of ‘s’ [-Wimplicit-int]
/tmp/code.c:1:6: warning: return type defaults to ‘int’ [-Wimplicit-int]
 s[9];main(j,a,t){for(;gets(s);putchar(a+48))for(j=t=strtol(s,a=0,2),j++;j--;)a+=j*j+j==t*2;}
      ^
/tmp/code.c: In function ‘main’:
/tmp/code.c:1:6: warning: type of ‘j’ defaults to ‘int’ [-Wimplicit-int]
/tmp/code.c:1:6: warning: type of ‘a’ defaults to ‘int’ [-Wimplicit-int]
/tmp/code.c:1:6: warning: type of ‘t’ defaults to ‘int’ [-Wimplicit-int]
/tmp/code.c:1:23: warning: implicit declaration of function ‘gets’ [-Wimplicit-function-declaration]
 s[9];main(j,a,t){for(;gets(s);putchar(a+48))for(j=t=strtol(s,a=0,2),j++;j--;)a+=j*j+j==t*2;}
                       ^
/tmp/code.c:1:31: warning: implicit declaration of function ‘putchar’ [-Wimplicit-function-declaration]
 s[9];main(j,a,t){for(;gets(s);putchar(a+48))for(j=t=strtol(s,a=0,2),j++;j--;)a+=j*j+j==t*2;}
                               ^
/tmp/code.c:1:53: warning: implicit declaration of function ‘strtol’ [-Wimplicit-function-declaration]
 s[9];main(j,a,t){for(;gets(s);putchar(a+48))for(j=t=strtol(s,a=0,2),j++;j--;)a+=j*j+j==t*2;}
                                                     ^
/tmp/cc08cLbM.o: In function `main':
code.c:(.text.startup+0xc): warning: the `gets' function is dangerous and should not be used.

strace