Wavey cuts
Geometry
Level
3
Consider the graph curves of 5th degree polynomials. What is the maximum number of enclosed regions that can be made from 10 of these curves?
The answer is 216.
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1 solution
Note that a new region is created for every two consecutive (in regards to the x-values) distinct intersections a new curve makes with any of the previous curves.
Note that the curves are 5th degree polynomials, thus they have pairwise at most 5 distinct intercepts. Thus, the nth curve can make at most 5(n-1) distinct intercepts with previous curves.
For k intersections ranked in ascending order of x values, there are k-1 pairs of consecutive intersections. Thus, the nth line forms at most 5(n-1)-1 = 5n-6 new regions.
Let f(a) denote the number of regions after the ath line is added, then: f(a) = f(a-1)+5n-6, with f(1)=0.
This implies that f(a) is effectively an arithmetic progression with starting term being f(2)=4. Using the summation formula shows that f(10) = 216.