Ημέρα 1η
1. The sequence
, 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?
2. Prove that the natural numbers can be divided into two groups in a way that both conditions are fulfilled:
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:
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 .
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
.
2. Let
be a quadratic trinomial. Given that the function
is increasing in the interval
, prove that:
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
.
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