1

Part of 1983 Bundeswettbewerb Mathematik

Problems(2)

soccer ball made up from black pentagons and white hexagons

Source: 1983 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p1

The surface of a soccer ball is made up of black pentagons and white hexagons together. On the sides of each pentagon are nothing but hexagons, while on the sides of each border of hexagons alternately pentagons and hexagons. Determine from this information about the soccer ball , the number of its pentagons and its hexagons.

3D geometrygeometryhexagonpentagon

trianglular billard

Source: 1983 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p1

The figure shows a triangular pool table with sides

a

b

c

. Located at point

S

c

a sphere - which can be assumed as a point. After kick-off, as indicated in the figure, it runs through as a result of reflections to

a, b, a, b

c

S

) always the same track. The reflection occurs according to law of reflection. Characterize entilrely all triangles

ABC

, which allow such an orbit, and determine the locus of

S

.
https://cdn.artofproblemsolving.com/attachments/5/b/7662943e5b9ad321226e0c5f5daa3c4ac9faaa.png

geometryreflectionLocus