MathDB

4

Part of ICMC 5

Problems(2)

Set closed under multiplication and reflection mod p

Source: ICMC 2022 Round 1 P4

12/4/2021
Let pp be a prime number. Find all subsets S⊆Z/pZS\subseteq\mathbb Z/p\mathbb Z such that 1. if a,b∈Sa,b\in S, then ab∈Sab\in S, and 2. there exists an r∈Sr\in S such that for all a∈Sa\in S, we have r−a∈S∪{0}r-a\in S\cup\{0\}.
Proposed by Harun Khan
reflectionGroupsnumber theorymod pcollege contestsICMC
Clean integers

Source: ICMC 2022 Round 2 P4

3/1/2022
Fix a set of integers SS. An integer is clean if it is the sum of distinct elements of SS in exactly one way, and dirty otherwise. Prove that the set of dirty numbers is either empty or infinite.
Note: We consider the empty sum to equal 00.
Proposed by Tony Wang and Ethan Tan
ICMCnumber theorycombinatorics