Ημέρα 1η
1. Triangle 
is right angled at 
. Lines 
and 
are internal angle bisectors. and intersect altitude 
at points 
and 
respectively. Prove that the line which passes through the midpoints of segments 
and 
is parallel to 
.
Κάντε κλικ εδώ, για να δείτε τη λύση της άσκησης, που μου έστειλε ο Νίκος Φραγκάκης, από την Ιεράπετρα.
is right angled at . Lines and are internal angle bisectors. and intersect altitude at points and respectively. Prove that the line which passes through the midpoints of segments and is parallel to .Κάντε κλικ εδώ, για να δείτε τη λύση της άσκησης, που μου έστειλε ο Νίκος Φραγκάκης, από την Ιεράπετρα.
2. The sequence 
is defined by
is defined by
and 
.
Prove that all terms of this sequence are perfect squares.
3. Prove that in the set consisting of 
people we can find a group of 
people in which everyone knows everyone or noone knows noone.
people we can find a group of people in which everyone knows everyone or noone knows noone. 
.
Find:
a) 
;
;
b) 
;
;
c) 
.
. 
Prove that the points 
and 
belong to Euler line of triangle 
where is circumcenter of .
and belong to Euler line of triangle where is circumcenter of .
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