MathDB

2

Part of STEMS 2023 Math Cat A

Problems(2)

Computational Set Construction

Source: STEMS 2023 Maths CAT A Part A P2

1/8/2023
Consider the set SS of permutations of 1,2,…,20221, 2, \dots, 2022 such that for all numbers kk in the permutation, the number of numbers less than kk that follow kk is even.
For example, for n=4;S={[3,4,1,2];[3,1,2,4];[1,2,3,4];[1,4,2,3]}n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}
If ∣S∣=(a!)b|S| = (a!)^b where a,b∈Na, b \in \mathbb{N}, then find the product abab.
combinatoricsSTEMS
STEMS 2023 Mathematics Category A Q2

Source:

1/8/2023
Given a complete bipartite graph on n,nn,n vertices (call this Kn,nK_{n,n}), we colour all its edges with 22 colours , red and blue . What is the least value of nn such that for any colouring of the edges of the graph , there will exist at least one monochromatic 44 cycle ?
graph theorycombinatorics