Tokarev S.I.

A set of five-digit numbers $\left\{N_{1}, N_{k}\right\}$ is such that any five-digit number, all digits of which are in non-decreasing order, coincides in at least one digit with at least one of the numbers $N_{1}, N_{k}$. Find the smallest possible value of $k$.
A set with the specified properties cannot consist of a single number. Indeed, for each $N=\overline{a b c d e}$, there is a number $G=\overline{\boldsymbol{g g 9 g g}}$ that differs from $N$ in all digits, where $g$ is a non-zero digit different from $a, b, c, d, e$. We will show that the numbers $N_{1}=13579$ and $N_{2}=12468$ form a set that satisfies the conditions of the problem.

Let $A=\overline{a_{1} a_{2} a_{3} a_{4} a_{5}}$ be an arbitrary number for which the digits satisfy the inequalities $1 \leq a_{1} \leq a_{2} \leq a_{3} \leq a_{4} \leq a_{5}$. Then, if $A$ does not match $N_{1}$ or $N_{2}$ in the units place, $a_{5} \leq 7$ and, consequently, $a_{4} \leq 7$; if there are no matches in the tens place either, then $a_{4} \leq 5$ and $a_{3} \leq 5$.

If, in addition, there are no matches in the hundreds place, then $a_{3} \leq 3$, from which it follows that $a_{2} \leq 3$; assuming further that $a_{2} \neq 2$ and $a_{2} \neq 3$, we arrive at the equality $a_{1}=1$, indicating that $A$ matches $N_{1}$ (and $N_{2}$) in the most significant digit.

## Answer

2.00