Theorems (in general sense) are constituents of inventing, analysing and solving olympiad tasks. Also, some theorems can be proved with computer assistance only. The main idea is (human) reducing of primary (unbounded) set to a finite one. Non-trivial immanent properties of mathematical objects are of interest because they can be considered as alternative definitions of these objects revealing their additional features. A non-formal indication of such property is only inital data (size of

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... ) and only output data (proven/not proven) in a corresponding algorithm. One new and two known examples of such properties are considered, some techniques to convert theorem-proving algorithms into olympiad tasks are proposed.