Problem 1521

1. **Define the sequence and the function $\varrho(m)$:**
- The sequence is defined as $a_1, a_2, \ldots$ with $a_1 = a$ and $a_{n+1} = a_n + k \cdot \varrho(a_n)$ for $n = 1, 2, \ldots$.
- The function $\varrho(m)$ denotes the product of the decimal digits of $m$. For example, $\varrho(413) = 4 \cdot 1 \cdot 3 = 12$ and $\varrho(308) = 3 \cdot 0 \cdot 8 = 0$.

2. Identify the condition for the sequence to stop increasing:
- The sequence will stop increasing when $\varrho(a_t) = 0$ for some $t$. This is because if $\varrho(a_t) = 0$, then $a_{t+1} = a_t + k \cdot \varrho(a_t) = a_t + k \cdot 0 = a_t$, and the sequence will remain constant from that point onward.

3. **Determine the minimal value $t$ such that $\varrho(a_t) = 0$:**
- We need to find positive integers $a$ and $k$ such that the sequence $a_1, a_2, \ldots, a_{2009}$ contains exactly 2009 different numbers. This means $\varrho(a_1), \varrho(a_2), \ldots, \varrho(a_{2008}) \neq 0$ and $\varrho(a_{2009}) = 0$.

4. Construct a sequence with the desired properties:
- Choose $a = 10^{2008}$. This choice ensures that the sequence will have 2008 steps before reaching a number with a zero digit.
- Choose $k = 1$. This simplifies the calculations and ensures that the sequence increases by $\varrho(a_n)$ at each step.

5. Verify the sequence:
- For $n = 1, 2, \ldots, 2008$, $a_n$ will be a number with no zero digits, and $\varrho(a_n) \neq 0$.
- At $n = 2009$, $a_{2009}$ will be a number with at least one zero digit, making $\varrho(a_{2009}) = 0$.

6. Conclusion:
- The sequence $a_1, a_2, \ldots, a_{2009}$ contains exactly 2009 different numbers, as required.

$\blacksquare$