MathDB

Problem 1

Part of 2011 VJIMC

Problems(2)

existence given P(1/n)=f(n)

Source: VJIMC 2011 1.1

6/1/2021
(a) Is there a polynomial P(x)P(x) with real coefficients such that P(1k)=k+2kP\left(\frac1k\right)=\frac{k+2}k for all positive integers kk? (b) Is there a polynomial P(x)P(x) with real coefficients such that P(1k)=12k+1P\left(\frac1k\right)=\frac1{2k+1} for all positive integers kk?
Polynomialsnumber theory
matrix product is nonzero

Source: VJIMC 2011 2.1

6/1/2021
Let n>kn>k and let A1,…,AkA_1,\ldots,A_k be real n×nn\times n matrices of rank n−1n-1. Prove that A1⋯Ak≠0.A_1\cdots A_k\ne0.
linear algebramatrix