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.

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According to Van der Waerden, the sum of two natural numbers is defined as &x*1=x& &x*y^+= x*y+x& (for every x and every y) The product of two natural numbers is recursively defined as &x*1=x& &x*y^+= x*y+x& (for every x and every y) x^+ ( y^+ ) here means the successor(consequent) of x( y) in the set of natural numbers.