8

Part of 2004 IMO Shortlist

Problems(2)

Cyclic quadrilateral geometry in the style of V. Thebault

Source: IMO Shortlist 2004 geometry problem G8

Given a cyclic quadrilateral

ABCD

M

be the midpoint of the side

CD

N

be a point on the circumcircle of triangle

ABM

. Assume that the point

N

is different from the point

M

\frac{AN}{BN}=\frac{AM}{BM}

. Prove that the points

E

F

N

are collinear, where

E=AC\cap BD

F=BC\cap DA

.
Proposed by Dusan Dukic, Serbia and Montenegro

geometrycircumcircleIMO Shortlistprojective geometryPolarsBrocardpower of a point

g(G)^3 <= c * f(G)^4

Source: IMO ShortList 2004, combinatorics problem 8

For a finite graph

G

f(G)

be the number of triangles and

g(G)

the number of tetrahedra formed by edges of

G

. Find the least constant

c

g(G)^3\le c\cdot f(G)^4

for every graph

G

.
Proposed by Marcin Kuczma, Poland

inequalitiesgraph theoryExtremal combinatoricsIMO Shortlist