1η Ημέρα
1. We have a line and 
points around it such that the distance of each point to the line is less than 
centimeters and the distance between any two points is more than 
centimeters. prove that there are two points such that their distance is at least 
meters.2. In triangle
, we have 
. perpendicular line from 
to side 
intersects angle bisector of 
in 
and perpendicular line from 
to side 
intersects angle bisector of 
in 
. prove that 
. 3. Find all increasing sequences
of natural numbers such that for each 
, number of the divisors of 
and 
is equal. (an increasing sequence is a sequence that if 
, then 
.)
2η Ημέρα
1. find the smallest natural number 
such that there exists real numbers in the interval 
such that their sum equals zero and the sum of their squares equals 
.
such that there exists real numbers in the interval such that their sum equals zero and the sum of their squares equals .2. rainbow is the name of a bird. this bird has colors and it's colors in two consecutive days are not equal. there doesn't exist 
days in this bird's life like 
such that 
and the bird has the same color in days 
and 
and the same color in days 
and 
different from the colors it has in days and . what is the maximum number of days rainbow can live in terms of ?
colors and it's colors in two consecutive days are not equal. there doesn't exist days in this bird's life like such that and the bird has the same color in days and and the same color in days and different from the colors it has in days and . what is the maximum number of days rainbow can live in terms of ?3. the line intersects the extension of in ( is nearer to than 
) and the extension of 
in ( is nearer to than ) of triangle . suppose that reflection of line to perpendicular bisector of side intersects the mentioned extensions in 
and 
respectively.prove that if 
, then 
.
intersects the extension of in ( is nearer to than ) and the extension of in ( is nearer to than ) of triangle . suppose that reflection of line to perpendicular bisector of side intersects the mentioned extensions in and respectively.prove that if , then . 
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