## IMO 1975

For students of class 6-8 (age 12 to 14)
umme.habiba.lamia
Posts: 2
Joined: Sun Oct 29, 2017 5:05 pm

### IMO 1975

Let f(n) denote the sum of the digits of n.

(a) For any integer n, prove that eventually the se quence f(n),f(f(n) ),f(f(f(n))), . . . will become constant. This constant value is called the digital sum of n.
(b) Prove that the digital sum of the product of any two twin primes, other than 3 and 5, is 8. (Twin primes are primes that are consecutive odd numbers, such as 17 and 19.)
(c) (IMO 1975) Let N = 4444^4444. Find f(f(f(n))), without a calculator.

samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

### Re: IMO 1975

(\$C\$)You can get the solution here

samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

### Re: IMO 1975

umme.habiba.lamia wrote:
Mon Dec 04, 2017 1:04 pm
Let \$f(n)\$ denote the sum of the digits of \$n\$.

(a) For any integer \$n\$, prove that eventually the se quence \$f(n),f(f(n) ),f(f(f(n))),\$ . . . will become constant. This constant value is called the digital sum of \$n\$.
(b) Prove that the digital sum of the product of any two twin primes, other than \$3\$ and \$5\$, is \$8\$. (Twin primes are primes that are consecutive odd numbers, such as \$17\$ and \$19\$.)
(c) (IMO 1975) Let \$N = 4444^4444.\$ Find \$f(f(f(n)))\$, without a calculator.
Problems are from The art and craft of problem solving book.