3. At present, the exchange rates of the US dollar and the euro are as follows: $D=6$ yuan and $E=7$ yuan. The People's Bank of China determines the yuan exchange rate independently of market conditions and adheres to a policy of approximate equality of currencies. One bank employee proposed the following scheme for changing the exchange rate to the management. Over one year, the exchange rates $D$ and $E$ are allowed to change according to the following four rules. Either change $D$ and $E$ to the pair $(D+E, 2D \pm 1)$, or to the pair $(D+E, 2E \pm 1)$. Moreover, it is forbidden for the exchange rates of the dollar and euro to be equal at any time.

For example: From the pair $(6,7)$, after one year, the following pairs can be made: $(13,11)$, $(11,13)$, $(13,15)$, or $(15,13)$. What is the smallest value that the difference between the larger and smaller of the resulting exchange rates can take after 101 years?
# Solution.

Note that if $D$ and $E$ have different parity, their sum is odd, and if they have the same parity, their sum is even. The second number in the pair is always odd. Initially, we have a pair $О Н$ (Odd, Odd).

After one year

After two years $\quad \mathrm{H}+\mathrm{H}=$ О, Н. And so on.

That is, after an odd number of years, there will be a pair of odd numbers.

Since they do not coincide, the smallest difference can only be 2.

We will also prove that after an even number of years, a difference of 1 can be achieved. We will prove this fact by mathematical induction.

For $n=1$, i.e., after one year $(13,11)$, the difference is 2. For $n=2(24,13 \cdot 2$ - $1=25)$. The difference is 1. Suppose for $n=2 k$ the difference is 1, i.e., $(2 m ; 2 m-1) \Rightarrow(4 m-$ $1 ; 4 m-3)$, the difference is 2. If $n=2 k-1$, the difference, by the induction hypothesis, is 2, then for $n=2 k$ from two courses $(2 m ; 2 m+2) \Rightarrow(4 m+2 ; 4 m+3)$. The difference is 1. Therefore, after 101 years, the smallest difference can be 2.

Answer: 2.

## Recommendations for checking.

Proved only the fact of the alternation of the parity of the difference of the courses - 2 points

Answer without justification - 0 points