Difference between revisions of "2017 UNCO Math Contest II Problems/Problem 6"
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== Problem == | == 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 == | == Solution == | ||
Revision as of 00:15, 20 May 2017
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]](http://latex.artofproblemsolving.com/e/0/e/e0e9809b75e06e64e84ac5f3f586c39872eeec32.png)
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
See also
| 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 | ||
