MathDB

2

Part of 2012 IMC

Problems(2)

IMC 2012 Day 1, Problem 2

Source:

7/28/2012
Let nn be a fixed positive integer. Determine the smallest possible rank of an n×nn\times n matrix that has zeros along the main diagonal and strictly positive real numbers off the main diagonal.
Proposed by Ilya Bogdanov and Grigoriy Chelnokov, MIPT, Moscow.
linear algebramatrixgeometryalgebrapolynomialinductionIMC
Recurrence and Series

Source: IMC 2012, Day 2, Problem 2

7/29/2012
Define the sequence a0,a1,…a_0,a_1,\dots inductively by a0=1a_0=1, a1=12a_1=\frac{1}{2}, and a_{n+1}=\dfrac{n a_n^2}{1+(n+1)a_n},   \forall n \ge 1. Show that the series ∑k=0∞ak+1ak\displaystyle \sum_{k=0}^\infty \dfrac{a_{k+1}}{a_k} converges and determine its value.
Proposed by Christophe Debry, KU Leuven, Belgium.
limitIMCcollege contests