MathDB

3

Part of 1990 China Team Selection Test

Problems(2)

Let's define an operation

Source: China TST 1990, problem 3

6/27/2005
In set SS, there is an operation ′′∘′′'' \circ '' such that ∀a,b∈S\forall a,b \in S, a unique a∘b∈Sa \circ b \in S exists. And (i) ∀a,b,c∈S\forall a,b,c \in S, (a∘b)∘c=a∘(b∘c)(a \circ b) \circ c = a \circ (b \circ c). (ii) a∘b≠b∘aa \circ b \neq b \circ a when a≠ba \neq b. Prove that: a.) ∀a,b,c∈S\forall a,b,c \in S, (a∘b)∘c=a∘c(a \circ b) \circ c = a \circ c. b.) If S={1,2,…,1990}S = \{1,2, \ldots, 1990\}, try to define an operation ′′∘′′'' \circ '' in SS with the above properties.
functionalgebra unsolvedalgebra
multiple of it with all digits not zero

Source: China TST 1990, problem 7

6/27/2005
Prove that for every integer power of 2, there exists a multiple of it with all digits (in decimal expression) not zero.
modular arithmeticinductionnumber theory unsolvednumber theory