3

Part of 1994 China Team Selection Test

Problems(2)

line of symmetry and every red point is reflected

Source: China TST 1994, problem 3

Find the smallest

n \in \mathbb{N}

such that if any 5 vertices of a regular

n

-gon are colored red, there exists a line of symmetry

l

n

-gon such that every red point is reflected across

l

to a non-red point.

symmetrygeometrygeometric transformationreflectiontrapezoidcombinatorics unsolvedcombinatorics

all the vertices of S are vertices of T

Source: China TST 1994, problem 6

For any 2 convex polygons

S

T

, if all the vertices of

S

are vertices of

T

S

a sub-polygon of

T

. I. Prove that for an odd number

n \geq 5

m

sub-polygons of a convex

n

-gon such that they do not share any edges, and every edge and diagonal of the

n

-gon are edges of the

m

sub-polygons. II. Find the smallest possible value of

m

combinatorics unsolvedcombinatorics