MathDB

Problem 1

Part of 2007 VJIMC

Problems(2)

two partitions of Q closed under addition/multiplication

Source: VJIMC 2007 1.1

6/24/2021
Can the set of positive rationals be split into two nonempty disjoint subsets Q1\mathbb Q_1 and Q2\mathbb Q_2, such that both are closed under addition, i.e. p+q∈Qkp+q\in\mathbb Q_k for every p,q∈Qkp,q\in\mathbb Q_k, k=1,2k=1,2? Can it be done when addition is exchanged for multiplication, i.e. p⋅q∈Qkp\cdot q\in\mathbb Q_k for every p,q∈Qkp,q\in\mathbb Q_k, k=1,2k=1,2?
advanced fields
construct A dense in [0,1]^2 and line intersections

Source: VJIMC 2007 2.1

6/24/2021
Construct a set A⊂[0,1]×[0,1]A\subset[0,1]\times[0,1] such that AA is dense in [0,1]×[0,1][0,1]\times[0,1] and every vertical and every horizontal line intersects AA in at most one point.
set theorydense