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ASU 260 All Soviet Union MO 1978 three automates and card pairs

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July 11, 2019
game strategycardscombinatorics

Problem Statement

Given three automates that deal with the cards with the pairs of natural numbers. The first, having got the card with (a,b)a,b), produces new card with (a+1,b+1)(a+1,b+1), the second, having got the card with (a,b)(a,b), produces new card with (a/2,b/2)(a/2,b/2), if both aa and bb are even and nothing in the opposite case; the third, having got the pair of cards with (a,b)(a,b) and (b,c)(b,c) produces new card with (a,c)(a,c). All the automates return the initial cards also. Suppose there was (5,19)(5,19) card initially. Is it possible to obtain
a) (1,50)(1,50)?
b) (1,100)(1,100)?
c) Suppose there was (a,b)(a,b) card initially (a<b)(a<b). We want to obtain (1,n)(1,n) card. For what nn is it possible?
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