IMO 1975(PROBLEM 4)

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Although your answer is correct, your solution is not complete

n (mod 9) does not always give the digit sum. For example if n = 2193 the digit sum is 2+1+9+3 which is 15 where as 2193 mod 9 is 6 when completely reduced. You must show bounds on the digit sum of digit sum of digit sum of 4444^4444 is less than 13 or even stronger bounds less than 13. Then it will be clear that the answer is true

farmer