International Math Olympiad Algebra Problem | aabb = x² | Find the Value of x 

Once we 99a +a+b is 11k^2 where a+b=11 9a+1=k^2 9a=k^2-1^2=(k+1)(k-1) If k+1=9,a=8-1=7 If 9=k-1,a=10+1=11,not a digit. So a=7,7+b=11,b=4 X^2=11(704)=11×11(64)=88×88 x=88

Ending digit of the square can be 0,1,4,5,6,9 b is not 0 or 1 Test for a=7 or 6 or 5 or 2. 11×704 or 11×605 or 11×506 or 11×209 I e 11×11×64,11×11×55,11×11×46,11×11×19 64 is only square with 11×11 a=7,b=4,x=88.

aabb=x^2 x=ab

9(7)+1=64 final answer

1^x=2 x=-Ln[Surd[2,2πn]]i

11^2(8^2)=(11)(11)(8)(8)=7744 7744=7744 solution