Ημέρα 1η
1. The sequence 
, consisting of natural numbers, is defined by the rule:
, consisting of natural numbers, is defined by the rule: 
.for every natural number 
, where 
is the number of the different divisors of (including 
and ). Is it possible that two consecutive members of the sequence are squares of natural numbers?
, where is the number of the different divisors of (including and ). Is it possible that two consecutive members of the sequence are squares of natural numbers? 2. Prove that the natural numbers can be divided into two groups in a way that both conditions are fulfilled:
For every prime number 
and every natural number , the numbers 
and 
do not have the same colour.
and every natural number , the numbers and do not have the same colour.There does not exist an infinite geometric sequence of natural numbers of the same colour.
3. We are given a real number 
, not equal to 
or . Sacho and Deni play the following game. First is Sasho and then Deni and so on (they take turns). On each turn, a player changes one of the “*” symbols in the equation:
, not equal to or . Sacho and Deni play the following game. First is Sasho and then Deni and so on (they take turns). On each turn, a player changes one of the “*” symbols in the equation:
with a number of the type 
, where is a whole number. Sasho wins if at the end the equation has no real roots, Deni wins otherwise. Determine (in term of ) who has a winning strategy .
, where is a whole number. Sasho wins if at the end the equation has no real roots, Deni wins otherwise. Determine (in term of ) who has a winning strategy .Ημέρα 2η
1. Let be an even natural number and let 
be the set of all non-zero sequences of length , consisting of numbers and (length binary sequences, except the zero sequence 
). Prove that can be partitioned into groups of three elements, so that for every triad 
, and for every 
, exactly zero or two of the numbers 
are equal to .
be an even natural number and let be the set of all non-zero sequences of length , consisting of numbers and (length binary sequences, except the zero sequence ). Prove that can be partitioned into groups of three elements, so that for every triad , and for every , exactly zero or two of the numbers are equal to . 2. Let 
be a quadratic trinomial. Given that the function 
is increasing in the interval 
, prove that:
be a quadratic trinomial. Given that the function is increasing in the interval , prove that:
for all real numbers 
such that 
and 
.
such that and . 3. We are given an acute-angled triangle 
and a random point 
in its interior, different from the centre of the circumcircle 
of the triangle. The lines 
and 
intersect for a second time in the points 
and 
respectively. Let 
and 
be the points that are symmetric of and in respect to 
and 
respectively. Prove that the circumcircle of the triangle and passes through a constant point that does not depend on the choice of .
and a random point in its interior, different from the centre of the circumcircle of the triangle. The lines and intersect for a second time in the points and respectively. Let and be the points that are symmetric of and in respect to and respectively. Prove that the circumcircle of the triangle and passes through a constant point that does not depend on the choice of .
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