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2

Part of 2011 IMC

Problems(2)

IMC 2011 Day 1 Problem 2

Source:

7/30/2011
Does there exist a real 3×33\times 3 matrix AA such that tr(A)=0\text{tr}(A)=0 and A2+At=I?A^2+A^t=I? (tr(A)\text{tr}(A) denotes the trace of A, AtA,\ A^t the transpose of A,A, and II is the identity matrix.)
Proposed by Moubinool Omarjee, Paris
linear algebramatrixalgebrapolynomialfunctionlinear algebra unsolved
IMC 2011 Day 2, Problem 2

Source:

8/1/2011
An alien race has three genders: male, female and emale. A married triple consists of three persons, one from each gender who all like each other. Any person is allowed to belong to at most one married triple. The feelings are always mutual ( if xx likes yy then yy likes xx). The race wants to colonize a planet and sends nn males, nn females and nn emales. Every expedition member likes at least kk persons of each of the two other genders. The problem is to create as many married triples so that the colony could grow.
a) Prove that if nn is even and k≥1/2k\geq 1/2 then there might be no married triple. b) Prove that if k≥3n/4k \geq 3n/4 then there can be formed nn married triple ( i.e. everybody is in a triple).
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