MathDB

3

Part of 2009 IMC

Problems(2)

IMC 2009 Day 1 P3

Source:

7/15/2014
In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from 11 to nn and let aia_i be the number of friends of the ithi^{\text{th}} resident. Suppose that ∑i=1nai2=n2−n \sum_{i=1}^{n}a_i^2=n^2-n Let kk be the smallest number of residents (at least three) who can be seated at a round table in such a way that any two neighbors are friends. Determine all possible values of k.k.
IMCcollege contests
IMC 2009 Day 2 P3

Source:

7/17/2014
Let A,B∈Mn(C)A,B\in \mathcal{M}_n(\mathbb{C}) be two n×nn \times n matrices such that A2B+BA2=2ABA A^2B+BA^2=2ABA Prove there exists k∈Nk\in \mathbb{N} such that (AB−BA)k=0n (AB-BA)^k=\mathbf{0}_n Here 0n\mathbf{0}_n is the null matrix of order nn.
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