MathDB

2

Part of 2005 ITAMO

Problems(2)

At least 1 among 18 divisible by the sum of its digits

Source: ItaMO 2005, P2

3/9/2012
Prove that among any 1818 consecutive positive integers not exceeding 20052005 there is at least one divisible by the sum of its digits.
number theory proposednumber theory
For which h, a_n=1

Source: ItaMO 2005, P5

3/9/2012
Let hh be a positive integer. The sequence ana_n is defined by a0=1a_0 = 1 and
an+1={an2 if an is even an+h otherwise .a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }\\\\a_n+h \text{ otherwise }.\end{array}
For example, h=27h = 27 yields a1=28,a2=14,a3=7,a4=34a_1=28, a_2 = 14, a_3 = 7, a_4 = 34 etc. For which hh is there an n>0n > 0 with an=1a_n = 1?
modular arithmeticEulerfunctionnumber theorytotient functionalgebra proposedalgebra