6

Part of 1998 IMO Shortlist

Problems(2)

geO 98 [convex hexagon ABCDEF with B + D + F = 360°]

Source: IMO Shortlist 1998 Geometry 6

ABCDEF

be a convex hexagon such that

\angle B+\angle D+\angle F=360^{\circ }

\frac{AB}{BC} \cdot \frac{CD}{DE} \cdot \frac{EF}{FA} = 1.

\frac{BC}{CA} \cdot \frac{AE}{EF} \cdot \frac{FD}{DB} = 1.

trigonometryanalytic geometrygeometryIMO Shortlistcomplex numbers

IMO ShortList 1998, combinatorics theory problem 6

Source: IMO ShortList 1998, combinatorics theory problem 6

Ten points are marked in the plane so that no three of them lie on a line. Each pair of points is connected with a segment. Each of these segments is painted with one of

k

colors, in such a way that for any

k

of the ten points, there are

k

segments each joining two of them and no two being painted with the same color. Determine all integers

k

1\leq k\leq 10

, for which this is possible.

combinatoricspoint setcombinatorial geometryColoringIMO Shortlist