1

Part of 2009 May Olympiad

Problems(2)

replacing number a by (1/2a) or (1-a)

Source: XV May Olympiad (Olimpiada de Mayo) 2009 L2 P1

Initially, the number

1

is written on the blackboard. At each step, the number on the blackboard is erased and another is written, which is obtained by applying any of the following operations:
Operation A: Multiply the number on the board with

\frac12

. Operation B: Subtract the number on the board from

1

.
For example, if the number

\frac38

is on the board, it can be replaced by

\frac12 \frac38=\frac{3}{16}

1-\frac38=\frac58

.
Give a sequence of steps after which the number on the board is

\frac{2009}{2^{20009}}

2-digit numbers that are assigned 8

Source: XV May Olympiad (Olimpiada de Mayo) 2009 L1 P1

Each two-digit natural number is assigned a digit as follows: Its digits are multiplied. If the result is a digit, this is the assigned digit. If the result is a two-digit number, these two figures are multiplied, and if the result is a digit, this is the assigned digit. Otherwise, the operation is repeated. For example, the digit assigned to

32

6

3 \times = 6

; the digit assigned to

93

4

9 \times 3 = 27

2 \times 7 = 14

1 \times 4 = 4

. Find all the two-digit numbers that are assigned

8

Digitsnumber theory