Is it possible for some natural number $k$ to divide all natural numbers from $1$ to $k$ into two groups and write down the numbers in each group in a row in some order so that you get two the same numbers?
HIDE: original wording beacuse it doesn't make much sense Можно ли при каком-то натуральном k разбить все натуральные числа от 1 до k на две группы и выписать числа в каждой группе подряд в некотором порядке так, чтобы получились два одинаковых числа?
"Is it possible, for some natural number ?, to split all natural numbers from 1 to ? into two groups and arrange the numbers in each group in some order such that the two resulting numbers are identical?"
Answer. Must not.
Decision. Let's assume the opposite. It is clear that $k > 10$, since there are no repeating ones in the set of digits from 1 to 9. Let us consider the largest power of ten $10^n$, which does not exceed $k$. The sequence of digits of the number $10^n$ will be included in one of the composed numbers in its entirety. But then the same sequence of one and $n$ subsequent zeros must be repeated in the second number. This sequence of digits could not have come from the union of two or more numbers (since natural numbers do not start with zeros), so it was contained in a single number other than $10^n$. But the smallest number other than $10^n$ that contains such a set of digits is $10^{n+1}$. We have obtained a contradiction with the fact that $10^n$ is the maximum power of ten, not exceeding $k$.