4. We will call a non-empty set of distinct natural numbers from 1 to 13 good if the sum of all the numbers in it is even. How many good sets are there in total?
4. Answer: $2^{12}-1$.

Solution. There are a total of $2^{13}$ different subsets (including the empty set) of 13 numbers. We will show that exactly half of them are good. Indeed, since the sum of all numbers from 1 to 13, which is 91, is an odd number, for each good subset of numbers, the remaining numbers form a bad subset, that is, a subset whose sum of numbers is an odd number. The reverse is also true, for each bad subset, the remaining numbers form a good subset. Since the subset of all numbers is bad, the corresponding good subset is the empty set, which does not satisfy the condition of the problem.

Criteria. Justified answer - 7 points; the empty set is considered - minus 1 point; answer without justification - 1 point.

Municipal stage of the All-Russian Schoolchildren's Olympiad in Mathematics Khanty-Mansi Autonomous District - Yugra

2019-2020 academic year