Ημέρα 1η
1. In triangle
it holds 
. Let 
and 
be midpoints of 
and 
, and let 
be the incenter of . Prove that 
are concyclic.
2. On semicircle, with diameter
, are given points 
and 
such that: 
and 
where 
are different positive integers. Find minimum possible value of 
.
3. Numbers 
are partitioned into two sequences 
and 
. Prove that number
is a perfect square.
Ημέρα 2η
1. Find maximum value of number
such that for any arrangement of numbers 
on a circle, we can find three consecutive numbers such their sum bigger or equal than .
2. Let
be positive reals such that 
. Prove that the inequality
holds.
3. In quadrilateral
sides 
and 
aren't parallel. Diagonals and 
intersect in 
. 
and 
are points on sides and 
such 
Prove that if 
are collinear then is cyclic.
1. In triangle
it holds . Let and be midpoints of and , and let be the incenter of . Prove that are concyclic. 2. On semicircle, with diameter
, are given points and such that: and where are different positive integers. Find minimum possible value of . are partitioned into two sequences and . Prove that numberis a perfect square.
Ημέρα 2η
1. Find maximum value of number
such that for any arrangement of numbers on a circle, we can find three consecutive numbers such their sum bigger or equal than . 2. Let
be positive reals such that . Prove that the inequality holds. 3. In quadrilateral
sides and aren't parallel. Diagonals and intersect in . and are points on sides and such Prove that if are collinear then is cyclic.
