1η ΗΜΕΡΑ
1. Let 
and 
be two positive integers. Consider a group of people such that, for each group of people, there is a 
-th person that knows them all (if 
knows 
then knows ).
and be two positive integers. Consider a group of people such that, for each group of people, there is a -th person that knows them all (if knows then knows ).1) If 
, prove that there exists a person who knows all others.
, prove that there exists a person who knows all others.2) If 
, give an example of such a group in which no-one knows all others.
, give an example of such a group in which no-one knows all others. 2. Let 
be an acute-angled triangle with 
. Let 
be the circumcircle, 
the orthocentre and 
the centre of . 
is the midpoint of 
. The line 
meets again at 
and the circle with diameter crosses again at 
. Prove that the lines 
are concurrent if and only if 
.
be an acute-angled triangle with . Let be the circumcircle, the orthocentre and the centre of . is the midpoint of . The line meets again at and the circle with diameter crosses again at . Prove that the lines are concurrent if and only if . 
2η ΗΜΕΡΑ
1. Let 
be an integer. A function 
is called -tastrophic when for every integer 
, we have 
where 
is the -th iteration of 
:
be an integer. A function is called -tastrophic when for every integer , we have where is the -th iteration of :
For which does there exist a -tastrophic function?
does there exist a -tastrophic function? 2. Determine all non-constant polynomials 
with integer coefficients for which the roots are exactly the numbers 
(with multiplicity).
with integer coefficients for which the roots are exactly the numbers (with multiplicity). 3. Let 
be a convex quadrilateral whose sides 
and are not parallel. Suppose that the circles with diameters 
and 
meet at points
and 
inside the quadrilateral. Let 
be the circle through the feet of the perpendiculars from to the lines 
and . Let 
be the circle through the feet of the perpendiculars from to the lines 
and . Prove that the midpoint of the segment 
lies on the line through of and .
be a convex quadrilateral whose sides and are not parallel. Suppose that the circles with diameters and meet at points and inside the quadrilateral. Let be the circle through the feet of the perpendiculars from to the lines and . Let be the circle through the feet of the perpendiculars from to the lines and . Prove that the midpoint of the segment lies on the line through of and .Proposed by Carlos Yuzo Shine, Brazil
Πηγή: artofproblemsolving
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