2017 UNCO Math Contest II Problems/Problem 6

Problem

$[asy] pair A=dir(72),B=dir(144),C=dir(216),D=dir(288),E=dir(360),O=(0,0); draw(A--B--C--D--E--A); pair AB1=(A+2*B)/3,AB2=(A+B)/2,AB3=(2*A+B)/3; draw(C--AB1--C--AB2--C--AB3); pair BC1=(B+2*C)/3,BC2=(B+C)/2,BC3=(2*B+C)/3; draw(D--BC1--D--BC2--D--BC3); pair CD1=(C+2*D)/3,CD2=(C+D)/2,CD3=(2*C+D)/3; draw(E--CD1--E--CD2--E--CD3); pair DE1=(D+2*E)/3,DE2=(D+E)/2,DE3=(2*D+E)/3; draw(A--DE1--A--DE2--A--DE3); pair EA1=(E+2*A)/3,EA2=(E+A)/2,EA3=(2*E+A)/3; draw(B--EA1--B--EA2--B--EA3); [/asy]$

The Spider's Divider

On a regular pentagon, a spider forms segments that connect one endpoint of each side to n different non-vertex points on the side adjacent to the other endpoint of that side, going around clockwise, as shown. Into how many non-overlapping regions do the segments divide the pentagon? Your answer should be a formula involving n. (In the diagram, n = 3 and the pentagon is divided into 61 regions.)

Solution

We can see that the web is made out of 5 congruent regions surrounding one central region. to find the number of parts in one of the five congruent regions, which is basically $n$ sections each divided by $n$ lines, so each section is divided into $n+1$ parts. The formula is thus $5n(n+1)+1$ or $\boxed{5n^2+5n+1}$

2017 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
All UNCO Math Contest Problems and Solutions

Page

Toolbox

Search

2017 UNCO Math Contest II Problems/Problem 6

Problem

Solution

See also