MathDB

2

Part of ICMC 4

Problems(2)

invertible matrix

Source: ICMC 2021 Round 1 P2

3/1/2022
Let AA be a square matrix with entries in the field Z/pZ\mathbb Z / p \mathbb Z such that An−IA^n - I is invertible for every positive integer nn. Prove that there exists a positive integer mm such that Am=0A^m = 0.
(A matrix having entries in the field Z/pZ\mathbb Z / p \mathbb Z means that two matrices are considered the same if each pair of corresponding entries differ by a multiple of pp.)
Proposed by Tony Wang
linear algebraMatricesNilpotentICMCcollege contestsmatrix
wonky sequence

Source: ICMC 2021 Round 2 P2

12/12/2021
Let p>3p > 3 be a prime number. A sequence of p−1p-1 integers a1,a2,…,ap−1a_1,a_2, \dots, a_{p-1} is called wonky if they are distinct modulo pp and aiai+2≢ai+12(modp)a_ia_{i+2} \not\equiv a_{i+1}^2 \pmod p for all i∈{1,2,…,p−1}i \in \{1, 2, \dots, p-1\}, where ap=a1a_p = a_1 and ap+1=a2a_{p+1} = a_2. Does there always exist a wonky sequence such that a1a2,a1a2+a2a3,…,a1a2+⋯+ap−1a1,a_1a_2, \qquad a_1a_2+a_2a_3, \qquad \dots, \qquad a_1a_2+\cdots +a_{p-1}a_1, are all distinct modulo pp?
Proposed by Harun Khan
number theoryICMCcollege contestsmod p