6.1 / 7.1 There are 8 rooks on the chessboard such that no two of them they don't hit each other. Prove that the black squares contain an even number of rooks.
6.2 The natural numbers are arranged in a 3×3 table. Kolya and Petya crossed out 4 numbers each. It turned out that the sum of the numbers crossed out by Petya is three times the sum numbers crossed out by Kolya. What number is left uncrossed?
\begin{tabular}{|c|c|c|}\hline 4 & 12 & 8 \\ \hline 13 & 24 & 14 \\ \hline 7 & 5 & 23 \\ \hline \end{tabular}
6.3 Misha and Sasha left at noon on bicycles from city A to city B. At the same time, I left from B to A Vanya. All three travel at constant but different speeds. At one o'clock Sasha was exactly in the middle between Misha and Vanya, and at half past one Vanya was in the middle between Misha and Sasha. When Misha will be exactly in the middle between Sasha and Vanya?
6.4 There are 35 piles of nuts on the table. Allowed to add one nut at a time to any 23 piles. Prove that by repeating this operation, you can equalize all the heaps.
6.5 There are 64 vertical stripes on the round drum, and each stripe you need to write down a six-digit number from digits 1 and 2 so that all the numbers were different and any two adjacent ones differed in exactly one discharge. How to do this?
6.6 / 7.6 Two brilliant mathematicians were told in natural terms number and were told that these numbers differ by one. After that they take turns asking each other the same question: “Do you know my number?" Prove that sooner or later one of them will answer positively.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here. leningrad math olympiadalgebrageometrycombinatoricsnumber theory