1η Ημέρα
1. Let 
be integer. Let 
, 
, ... 
be sequence of positive reals such that:
, for 
.Prove
. 2. Let
be an odd positive integer such that both 
and 
are powers of two. Prove 
is power of two or 
.
3. Let
be orthocenter and 
circumcenter of an acuted angled triangle 
. 
and 
are feets of perpendiculars from 
and 
on 
and 
respectively. Let 
and 
intersect 
and 
in 
and 
, respectively. Let 
be intersection of circumcircles of 
and 
different than , and 
is midpoint of 
. Prove that 
are collinear iff is circumcenter of 
.2η Ημέρα
1. On sides 
are points 
, respectively, such that 
; 
. , are midpoints of 
and 
. is circumcenter of , 
, 
are symmetric with with respect to and . Prove that 
are concyclic. 2. Are there positive integers
greater than 
such that:
? 3. Set
consists of 
points in plane, and 
consists of 
lines in plane. Pair 
is good if 
, 
and 
. Prove that maximum number of good pairs is no greater than 
, and prove that there exits configuration with exactly good pairs.
