ΗΜΕΡΑ 1η
1 Prove whether or not there exist natural numbers 
where 
such that
2 Let
be a polynomial of degree 
with the leading coefficient positive and 
for 
Prove that if the equation 
has four different non-positive real roots, then for arbitrary 
then 
has 
different real roots. 
3 Triangle 
and a function 
have the following property: for every line segment 
from the interior of the triangle with midpoint 
, the inequality 
, where 
is the distance from point 
to the nearest side of the triangle ( is in the interior of 
). Prove that for each line segment 
and each point interior point 
the inequality 
holds.
and a function have the following property: for every line segment from the interior of the triangle with midpoint , the inequality , where is the distance from point to the nearest side of the triangle ( is in the interior of ). Prove that for each line segment and each point interior point the inequality holds. 1 Point 
is inside . The feet of perpendicular from to 
are 
. Perpendiculars from 
and 
respectively to 
and 
meet at 
. Let 
be the foot of perpendicular from to 
. Prove that 
are concyclic.
2 For each natural number
we denote 
and 
the number of natural numbers dividing and the number of natural numbers less than that are relatively prime to . Find all natural numbers for which has exactly two different prime divisors and satisfies 
.
3 In the interior of the convex 2011-gon are
points, such that no three among the given 
points (the interior points and the vertices) are collinear. The points are coloured one of two different colours and a colouring is called "good" if some of the points can be joined in such a way that the following conditions are satisfied:
1) Each segment joins two points of the same colour.
2) None of the line segments intersect.
3) For any two points of the same colour there exists a path of segments connecting them.
Find the number of "good" colourings.
is inside . The feet of perpendicular from to are . Perpendiculars from and respectively to and meet at . Let be the foot of perpendicular from to . Prove that are concyclic. 2 For each natural number
we denote and the number of natural numbers dividing and the number of natural numbers less than that are relatively prime to . Find all natural numbers for which has exactly two different prime divisors and satisfies . 3 In the interior of the convex 2011-gon are
points, such that no three among the given points (the interior points and the vertices) are collinear. The points are coloured one of two different colours and a colouring is called "good" if some of the points can be joined in such a way that the following conditions are satisfied:1) Each segment joins two points of the same colour.
2) None of the line segments intersect.
3) For any two points of the same colour there exists a path of segments connecting them.
Find the number of "good" colourings.
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου