2011/030) prove that

(1....1) = (22....2)+ (3333....3)^2
2n times ntimes + n times

Proof:

LHS = (99999...9)/9 = (10^2n -1 )/ 9
RHS = 2/9( 10^n-1)+ (10^n-1)/9

so we need to prove that (RHS = LHS as RHS is more complex)

(10^n-1 )^2 + 2(10^n-1) = (10^2n- 1)

LHS = 10^2n - 2* 10^n +1 + 2*10^n - 2 = (10^2n- 1)