Subcontests
(4)a³,b³,a+b rational, but a,b not rational
a) Give example of two irrational numbers a,b having the property that a3,b3,a+b are all rational.b) Prove that if x,y are two nonnegative real numbers having the property that x3,y3,x+y are rational, then x and y are both rational.
Mihai Piticari and Vladimir Cerbu Linear algebra impossibility
Do exist pairwise distinct matrices A,B,C∈M2(R) verifying the following properties?(i)detA=detC
(ii)AB=C,BC=A,CA=B
(iii)trA,trB=0
Robert Pop