Some non-standard logic problems

Participants in the competition are asked to solve a system of three equations with three unknowns of the form $A_1=ab+c$, $A_2=ac+b$, $A_3=bc+a$, where $a$, $b$, $c$ — unknowns taking natural values greater than 1. The participants know only one of the numbers on the left sides of the equations, $A_1$, $A_2$, $A_3$, as well as the results of the survey of participants by the presenter. Five problems are formulated and solved with a complete analysis of the solution. Algorithms for solving these problems are proposed.