IMO 1986 Problem 3

The problem is less interesting once we set out to find a quantity that strictly increases or decreases while keeping something else a constant. It just felt more like an algebraic problem.

To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

Solution

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Yes, it does.

Let the numbers be $a_1,\dots,a_5$, and define $a_6:=a_1$. It could be checked that when we operate on $a_j$, the total sum does not change while the quantity $$J(a_1,\dots,a_5):=\sum_{i=1}^5a_i^2+(a_i+a_{i+1})^2$$ changes by $$2a_j(a_1+\dots+a_5)<0.$$ Since $J(a_1,\dots,a_5)$ is nonnegative, the process necessarily terminates after any finite number of steps. The statement holds even when the numbers are not integral.