UKMT Special (Problem 2222)

Each of AA and BB is a 44-digit palindromic integer.

CC is a 33-digit palindromic integer and

AB=CA - B = C

What are the possible values of CC?

[UKMT Hamilton Olympiad 20192019, H44]

Note by Yajat Shamji
8 months, 2 weeks ago

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You have until next Saturday, 3:003:00pm!

Yajat Shamji - 8 months, 2 weeks ago

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Rewrite as

   a b b a
   c d d c
   _______-
     e f e

Step 1) Notice that the difference of 2 four digit number is a three digit number, this implies that (look at thousand column) a - c = 0 or a - c - 1 = 0 (in the case of hundred column borrowing from thousands column).

   If a - c = 0, then (look at ones column) e = 0 
   and implies that the subtraction result is a two digit number which is not true. 
   Therefore, a - c - 1 = 0   ->   a = c + 1.

Step 2) Notice that (look at ones column) a - c = e or 10 + a - c = e (in the case of hundred column borrowing from tens column).

   Substitute a from step 1.
   If 10 + a - c = e, then 10 + a - c = e     ->     10 + c + 1 - c = e     ->     e = 11 
   However, c is a one digit, so this assumption must be false.
   Therefore, a - c = e     ->     c + 1 - c = e     ->     e = 1

   a b b a
   c d d c
   _______-
     1 f 1

Step 3) Notice that a = c + 1, however, (look at thousands column) the result is three digit, this implies that b < d and borrow from the thousands column , 10 + b - d = 1 or 10 + b - d - 1 = 1

   If 10 + b - d = 1, this implies that the tens column doesn't borrow, however, b < d, 
   so this assumption must be false.
   Therefore, 10 + b - d - 1 = 1     ->     8 + b = d       ->     b = 0, d = 8  or  b = 1, d = 9
   Otherwise, d is a two digit number

Step 4) Notice that, since a = c + 1 the ones column doesn't borrow from the tens column and b < d, therefore 10 + b - d = f (in the case of hundred column borrowing from tens column).

   Substitute b and d from step 3.
   Case 1 : b = 0, d = 8 
   10 + 0 - 8 = f     ->     f = 2
   Case 2 : b = 1, d = 9
   10 + 1 - 9 = f     ->     f = 2
   In either case, f = 2

   a b b a
   c d d c
   _______-
     1 2 1

Therefore, C = 121

Quest Keeper - 8 months, 2 weeks ago

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That's wrong. Try again!

Yajat Shamji - 8 months, 2 weeks ago

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I see, I forgot the carrier. BTW, do you do integral?

Quest Keeper - 8 months, 1 week ago

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@Quest Keeper No.

Yajat Shamji - 8 months, 1 week ago

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