MathDB

2

Part of 2014 IMC

Problems(2)

IMC 2014, Problem 2

Source: IMC 2014

7/27/2016
Consider the following sequence (an)n=1∞=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,… )(a_n)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots) Find all pairs (α,β)(\alpha, \beta) of positive real numbers such that lim⁡n→∞∑k=1naknα=β\lim_{n\to \infty}\frac{\displaystyle\sum_{k=1}^n a_k}{n^{\alpha}}=\beta.
(Proposed by Tomas Barta, Charles University, Prague)
IMCSequencescollege contests
IMC 2014, Problem 7

Source: IMC 2014

7/27/2016
Let A=(aij)i,j=1nA=(a_{ij})_{i, j=1}^n be a symmetric n×nn\times n matrix with real entries, and let λ1,λ2,…,λn\lambda _1, \lambda _2, \dots, \lambda _n denote its eigenvalues. Show that ∑1≤i<j≤naiiajj≥∑1≤i<j≤nλiλj\sum_{1\le i<j\le n} a_{ii}a_{jj}\ge \sum_{1\le i < j\le n} \lambda _i \lambda _j and determine all matrices for which equality holds.
(Proposed by Matrin Niepel, Comenius University, Bratislava)
IMCcollege contestslinear algebramatrixsuperior algebra