Problem 3

To each vertex of a regular pentagon an integer is assigned, with positive total sum. If three consecutive vertices carry

x,y,z

and

y<0

, replace them by

x+y,-y,z+y

. This operation is repeated whenever some number is negative. Must the procedure always end after finitely many steps?

Step 1 of 4: Preserve the positive sum

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In plain words

The move redistributes the three selected entries but does not change their sum, so the global sum remains a positive invariant.

S=x_1+x_2+x_3+x_4+x_5>0

Detailed analysis

Write the five values cyclically as $(x_1,x_2,x_3,x_4,x_5)$ . Each move replaces $x,y,z$ by $x+y,-y,z+y$ , whose sum is still $x+y+z$ . Thus $S=\sum_{i=1}^5x_i$ is unchanged and positive.