MathDB

Problem 1

Part of 2003 VJIMC

Problems(2)

d(n^2+1) is not monotone for n>n_0, any n_0

Source: VJIMC 2003 1.1

7/14/2021
Let d(k)d(k) denote the number of natural divisors of a natural number kk. Prove that for any natural number n0n_0 the sequence {d(n2+1)}n=n0∞\left\{d(n^2+1)\right\}^\infty_{n=n_0} is not strictly monotone.
number theory
A^2002=B^2003=I and AB=BA implies A+B+I invertible

Source: VJIMC 2003 2.1

7/13/2021
Two real square matrices AA and BB satisfy the conditions A2002=B2003=IA^{2002}=B^{2003}=I and AB=BAAB=BA. Prove that A+B+IA+B+I is invertible. (The symbol II denotes the identity matrix.)
matrixlinear algebra