MathDB

2

Part of 2023 Czech-Polish-Slovak Junior Match

Problems(2)

number of positive divisors of n is second largest divisor of n.

Source: 2023 Czech-Polish-Slovak Match Junior, individual p2 CPSJ

5/5/2024
For a positive integer nn, let d(n)d(n) denote the number of positive divisors of nn. Determine all positive integers nn for which d(n)d(n) is the second largest divisor of nn.
number theorynumber of divisors
permutations of numbers 1-2023

Source: 2023 Czech-Polish-Slovak Match Junior, Team p2 CPSJ

5/5/2024
The numbers 1,2,...,20231, 2,..., 2023 are written on the board in this order. We can repeatedly perform the following operation with them: We select any odd number of consecutively written numbers and write these numbers in reverse order. How many different orders of these 20232023 numbers can we get?
Example: If we start with only the numbers 1,2,3,4,5,61, 2, 3, 4, 5, 6, we can perform the following steps 1,2,3,4,5,6→3,2,1,4,5,6→3,6,5,4,1,2→3,6,1,4,5,2→...1, 2, 3, 4, 5, 6 \to 3, 2, 1,4, 5, 6 \to 3, 6, 5, 4, 1, 2 \to 3, 6, 1, 4, 5, 2 \to ...
combinatorics