Ημέρα 1η
1. Consider a circle 
and a point 
on it. Circle 
with center , intersects in two points 
and 
. 
is a circle which is externally tangent to at 
and internally tangent to at 
and suppose that 
passes through . Suppose 
and 
are second intersection points of 
and 
with . Prove that 
is parallel with 
.
and a point on it. Circle with center , intersects in two points and . is a circle which is externally tangent to at and internally tangent to at and suppose that passes through . Suppose and are second intersection points of and with . Prove that is parallel with . 
2. Suppose 
is a natural number. In how many ways can we place numbers 
around a circle such that each number is a divisor of the sum of it's two adjacent numbers?
is a natural number. In how many ways can we place numbers around a circle such that each number is a divisor of the sum of it's two adjacent numbers? 3. Prove that if 
is a natural number then there exists a natural number 
such that 
and none of the numbers 
are perfect powers.
is a natural number then there exists a natural number such that and none of the numbers are perfect powers. 1. a) Do there exist 
-element subsets 
of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number , sum of the elements of 
equals 
?
-element subsets of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number , sum of the elements of equals ?b) Do there exist -element subsets of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number , sum of the elements of equals 
?
-element subsets of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number , sum of the elements of equals ? 2. Consider the second degree polynomial 
with real coefficients. We know that the necessary and sufficient condition for this polynomial to have roots in real numbers is that its discriminant, 
be greater than or equal to zero. Note that the discriminant is also a polynomial with variables 
and 
. Prove that the same story is not true for polynomials of degree 
: Prove that there does not exist a variable polynomial 
such that:
with real coefficients. We know that the necessary and sufficient condition for this polynomial to have roots in real numbers is that its discriminant, be greater than or equal to zero. Note that the discriminant is also a polynomial with variables and . Prove that the same story is not true for polynomials of degree : Prove that there does not exist a variable polynomial such that:The fourth degree polynomial 
can be written as the product of four 
st degree polynomials if and only if 
. (All the coefficients are real numbers.)
can be written as the product of four st degree polynomials if and only if . (All the coefficients are real numbers.) 3. The incircle of triangle 
, is tangent to sides 
and 
in 
and 
respectively. The reflection of with respect to 
and the reflection of 
with respect to 
are 
and respectively. Prove that the incenter of triangle 
is inside or on the incircle of triangle .
, is tangent to sides and in and respectively. The reflection of with respect to and the reflection of with respect to are and respectively. Prove that the incenter of triangle is inside or on the incircle of triangle . Κάντε κλικ εδώ, για να τα κατεβάσετε.
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