MathDB

4

Part of 2015 China Second Round Olympiad

Problems(2)

Find k not satisfying divisibility condition

Source: China Second Round 2015 (A) Q4

5/5/2016
Find all positive integers kk such that for any positive integer nn, 2(k−1)n+12^{(k-1)n+1} does not divide (kn)!n!\frac{(kn)!}{n!}.
number theorylegendre s theoremChina second round
Number of good sequences

Source: China Second Round 2015 (B) Q4

5/5/2016
Given positive integers m,n(2≤m≤n)m,n(2\le m\le n), let a1,a2,…,ama_1,a_2,\ldots ,a_m be a permutation of any mm pairwise distinct numbers taken from 1,2,…,n1,2,\ldots ,n. If there exist k∈{1,2,…,m}k\in\{1,2,\ldots ,m\} such that ak+ka_k+k is odd, or there exist positive integers k,l(1≤k<l≤m)k,l(1\le k<l\le m) such that ak>ala_k>a_l, then call a1,a2,…,ama_1,a_2,\ldots ,a_m a good sequence. Find the number of good sequences.
combinatorics