2

Part of 2007 Iran MO (2nd Round)

Problems(2)

Prove that PE/PF=ME/NF - Iran NMO 2007 - Problem5

Source:

C,D

are exterior tangent to each other at point

P

A

is in the circle

C

2

AM,AN

A

D

M,N

are the tangency points.). The second meet points of

AM,AN

C

E,F

, respectively. Prove that

\frac{PE}{PF}=\frac{ME}{NF}

geometrygeometry proposed

Vertices of a cube - Iran NMO 2007 - Problem2

Source:

Two vertices of a cube are

A,O

AO

is the diagonal of one its faces. A

n-

run is a sequence of

n+1

vertices of the cube such that each

2

consecutive vertices in the sequence are

2

ends of one side of the cube. Is the

1386-

O

to itself less than

1386-

O

A

or more than it?

geometry3D geometrycombinatorics unsolvedcombinatorics