In this paper we extend and deepen a shortlist for the 37th International Mathematical Olympiad (IMO) and propose the Frog Leap Commute Theorem and the Queue Polynomial.
We explore the problem from the following aspects:
(1) Make use of semi-invariants and propose the Frog Leap Commute Theorem.
(2) Make extensions regarding frogs leaping to opposite directions on a straight line.
(3) Research frogs leaping to the same direction on a straight line and solve the minimum number of frogs satisfying an infinite leap.
(4) Extend the problem to leaps on a plane or in space.
(5) Research and extend problems regarding frogs leaping on a circle.
(6) Estimate the function c(n) and calculate the order of the function.