# 2015 UNCO Math Contest II Problems/Problem 5

## Problem

$[asy] pair a1=(0,0),b1=(1,0),c1=(3,0),d1=(4,0),e1=(4.5,-.25),f1=(4.25,-2),g1=(3.75,-2.25),h1=(1.5,-2); pair i1=(.75,-3),j1=(1.25,-3),k1=(3.25,-3),l1=(4,-2.75); draw(a1--b1--c1--d1--e1--f1--d1--g1--f1,dot); draw(b1--h1--c1--g1--h1,dot); draw(a1--i1--j1--h1,dot); draw(j1--k1--l1--f1,dot); draw(k1--g1,dot); [/asy]$

A termite nest has the shape of an irregular polyhedron. The bottom face is a quadrilateral. The top face is another polygon. The sides comprise $9$ triangles, $6$ quadrilaterals, and $1$ pentagon. The nest has $10$ vertices on its sides and bottom, not counting the several around the top face. How many edges does the top face have?

You may use Euler’s polyhedral identity, which says that on a convex polyhedron the number of faces plus the number of vertices is two more than the number of edges. (A vertex is a corner point and an edge is a line segment along which two faces meet.)

## Solution

$8$

## See also

 2015 UNCO Math Contest II (Problems • Answer Key • Resources) Preceded byProblem 4 Followed byProblem 6 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 All UNCO Math Contest Problems and Solutions