If a sequence $\left\{a_{1}, a_{2}, \ldots, a_{n}\right\}$ of positive integers (where $n$ is a positive integer) has the property that the last digit of $a_{k}$ is the same as the first digit of $a_{k+1}$ (here $k=1,2, \ldots, n$ and we define $\left.a_{n+1}=a_{1}\right)$, then the sequence is said to be a 'dragon sequence'. For example, $\{414\},\{208,82\}$ and $\{1,17,73,321\}$ are all 'dragon sequences'. At least how many two-digit numbers must be chosen at random to ensure that a 'dragon sequence' can be formed among some of the chosen numbers?
(2 marks)
If a sequence $\left\{a_{1}, a_{2}, \ldots, a_{n}\right\}$ (where $n$ is a positive integer) of positive integers has the property that the last digit of $a_{k}$ is the same as the first digit of $a_{k+1}$ (here $k=1,2, \ldots, n$, and we define $a_{n+1}=a_{1}$), then the sequence is called a 'dragon sequence'. For example, \{414\}, \{208,82\}, and $\{1,17,73,321\}$ are all 'dragon sequences'. At least how many two-digit numbers must be chosen at random to ensure that a 'dragon sequence' can be formed among some of the chosen numbers?
(2 marks)
11. 46
11. If we take any 46 two-digit numbers, then only 44 two-digit numbers are not chosen. Hence, we either have one of the 36 pairs $\{\overline{A B}, \overline{B A}\}$ where $1 \leq A < B \leq 9$, or we have a sequence of 45 numbers where $A_{1} < A_{2} < \cdots < A_{n-1} < A_{n}$ or $A_{n} > A_{n-1} > \cdots > A_{2} > A_{1}$, which is a contradiction. So these 45 numbers contain no 'dragon sequence'. It follows that the answer is 46.