1. Find the smallest natural number $k$, for which the following holds: If we select any $k$ different numbers from the set $\{1,4,7,10,13, \ldots, 1999\}$, then there exist two different numbers among the selected ones whose sum equals 2000.
1. Let $\mathrm{M}=\{1,4,7,10,13, \ldots, 1999\}$ and list all sums of two different (we will not emphasize this further) numbers from M that equal the number 2000:

$$
2000=1+1999=4+1996=7+1993=\ldots=997+1003 \text {. }
$$

With the exception of the single number 1000, each number in M appears in exactly one sum (the sum $1000+1000$ is not considered according to the problem statement). Since $997=1+3 \cdot 332$, exactly 333 sums are listed. Therefore, it is possible to select 334 numbers from M (one from each listed sum, along with the number 1000, for example, the numbers $1,4,7, \ldots, 997,1000$) such that the sum of any two selected numbers is not 2000. However, if we select any 335 numbers from M, then some two of the selected numbers are the addends of one of the listed sums (let's repeat: there are 333 listed sums, and only one number from $\mathrm{M}$ is missing from them). Therefore, $k=335$ is the desired value.

For a complete solution, 6 points are awarded: 3 points for explaining that the number $k=334$ does not satisfy the condition, and 3 points for justifying that the number $k=335$ does satisfy the condition.