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May 18, 2020

Convex polyhedron Problem | AIME I, 1988 | Question 10

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1988 based on convex polyhedron.

Convex polyhedron Problem - AIME I, 1988


A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face?

  • is 107
  • is 840
  • is 634
  • cannot be determined from the given information

Key Concepts


Integers

Edges

Algebra

Check the Answer


Answer: is 840.

AIME I, 1988, Question 10

Geometry Revisited by Coxeter

Try with Hints


First hint

\({48 \choose 2}\)=1128

Every vertex lies on exactly one vertex of a square/hexagon/octagon

V=(12)(4)=(8)(6)=(6)(8)=48

Second Hint

each vertex is formed by the trisection of three edges and every edge is counted twice, once at each of its endpoints, the number of edges E=\(\frac{3V}{2}\)=72

Final Step

each of the segment on face of polyhedron is diagonal of that face, so each square gives \(\frac{n(n-3)}{2}=2\) diagonals, each hexagon=9,each octagon=20. The number of diagonals is \((2)(12)+(9)(8)+(20)(6)\)=216

or, number of space diagonals =1128-72-216=840.

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