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PrintBelarusian Mathematical Olympiad
Belarus algebra
Problem
On planet Automoria there live a (possible infinite) number Automorians. Any Automorian have feelings like love and respect. It is known that 1) each Automorian loves exactly one Automorian and respects exactly one Automorian; 2) if loves , then every Automorian respecting also loves ; 3) if respects , then every Automorian loving also respects ; 4) for every Automorian there is somebody loving him. Is it true that every Automorian respects the Automorian he loves?
Solution
Answer: yes. Denote by and the Automorians whom the Automorian loves and respects, respectively. The first condition of the problem implies that and are well defined functions on the set of Automorians. We have to prove that these functions are equal.
Take any Automorian , then he respects , who in turn loves . Using the second condition of the problem we see that this Automorian is also loved by , thus Similarly we get from the third condition that The fourth condition implies that the function must be surjective.
Equation (2) gives and using (1) we obtain . Since is surjective, we must have for every Automorian . Then (1) now gives and hence .
Thus, every Automorian loves and respects himself only.
Take any Automorian , then he respects , who in turn loves . Using the second condition of the problem we see that this Automorian is also loved by , thus Similarly we get from the third condition that The fourth condition implies that the function must be surjective.
Equation (2) gives and using (1) we obtain . Since is surjective, we must have for every Automorian . Then (1) now gives and hence .
Thus, every Automorian loves and respects himself only.
Final answer
yes
Techniques
Injectivity / surjectivityExistential quantifiersFunctional equations