Computing 999,999⋅222,222+333,333⋅333,334999,999\cdot 222,222 + 333,333\cdot 333,334 by hand.

I got this question from a last year’s olympiad paper.

Compute 999,999222,222+333,333333,334.

Is there an approach to this by using pen-and-paper?

EDIT Working through on paper made me figure out the answer. Posted below.

I’d now like to see other methods. Thank you.

Answer

My observation suggests that we may simplify it somewhat like this assuming x=111,111:
9x2x+3x(3x+1)=9x2+18x2+3x=27x2+3x=3x(9x+1)

So this means:
3x(9x+1)=333,333(999,999+1)=333,3331,000,000=333,333,000,000

Attribution
Source : Link , Question Author : P.K. , Answer Author : OrangeDog

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