MathDB

5

Part of 2021 South East Mathematical Olympiad

Problems(2)

Number of a sequence

Source: 2021China South East Mathematical Olympiad Grade 10 P5

7/29/2021
To commemorate the 43rd43rd anniversary of the restoration of mathematics competitions, a mathematics enthusiast arranges the first 20212021 integers 1,2,…,20211,2,\dots,2021 into a sequence {an}\{a_n\} in a certain order, so that the sum of any consecutive 4343 items in the sequence is a multiple of 4343. (1) If the sequence of numbers is connected end to end into a circle, prove that the sum of any consecutive 4343 items on the circle is also a multiple of 4343; (2) Determine the number of sequences {an}\{a_n\} that meets the conditions of the question.
Sequencenumber theory
China South East Mathematical Olympiad 2021 Grade11 P5

Source:

8/14/2021
Let A={a1,a2,⋯ ,an,b1,b2,⋯ ,bn}A=\{a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n\} be a set with 2n2n distinct elements, and Bi⊆AB_i\subseteq A for any i=1,2,⋯ ,m.i=1,2,\cdots,m. If ⋃i=1mBi=A,\bigcup_{i=1}^m B_i=A, we say that the ordered m−m-tuple (B1,B2,⋯ ,Bm)(B_1,B_2,\cdots,B_m) is an ordered m−m-covering of A.A. If (B1,B2,⋯ ,Bm)(B_1,B_2,\cdots,B_m) is an ordered m−m-covering of A,A, and for any i=1,2,⋯ ,mi=1,2,\cdots,m and any j=1,2,⋯ ,n,j=1,2,\cdots,n, {aj,bj}\{a_j,b_j\} is not a subset of Bi,B_i, then we say that ordered m−m-tuple (B1,B2,⋯ ,Bm)(B_1,B_2,\cdots,B_m) is an ordered m−m-covering of AA without pairs. Define a(m,n)a(m,n) as the number of the ordered m−m-coverings of A,A, and b(m,n)b(m,n) as the number of the ordered m−m-coverings of AA without pairs. (1)(1) Calculate a(m,n)a(m,n) and b(m,n).b(m,n). (2)(2) Let m≥2,m\geq2, and there is at least one positive integer n,n, such that a(m,n)b(m,n)≤2021,\dfrac{a(m,n)}{b(m,n)}\leq2021, Determine the greatest possible values of m.m.
combinatoricsset