4

Part of 1985 Iran MO (2nd round)

Problems(2)

At least one of x or y is integer - [Iran Second Round 1985]

Source:

x

y

be two real numbers. Prove that the equations \lfloor x \rfloor + \lfloor y \rfloor =\lfloor x +y \rfloor , \lfloor -x \rfloor + \lfloor -y \rfloor =\lfloor -x-y \rfloor Holds if and only if at least one of

x

y

floor functionnumber theory proposednumber theory

G_a is a subgroup of G - [Iran Second Round 1985]

Source:

G

be a group and let

a

be a constant member of it. Prove that

G_a = \{x | \exists n \in \mathbb Z , x=a^n\}

Is a subgroup of

G.

group theoryabstract algebrasuperior algebrasuperior algebra unsolved