Ημέρα 1η
1. Determine if the number
is irrational for all non-negative integers 
.
. 2. Find all functions 
such that
such that
for all 
.
. 3. Let 
and be integers such that 
. We have 
separate boxes and 
identical balls. Define 
as the number of ways that the balls can be distributed into the boxes. Let 
be a prime number. Find the integers greater than 
such that the prime number is a divisor of for all 
.
and be integers such that . We have separate boxes and identical balls. Define as the number of ways that the balls can be distributed into the boxes. Let be a prime number. Find the integers greater than such that the prime number is a divisor of for all . Ημέρα 2η
. 2. A sequence 
of integers is defined by the recurrence
of integers is defined by the recurrence
Prove that all terms of the sequence are integers and find an explicit formula for 
.
. 3. Let 
be an acute-angled triangle. Let 
be the inscribed circle with centre 
, 
be the circumscribed circle with centre 
and 
be the midpoint of the altitude 
where 
lies on 
. The circle be tangent to the side at the point 
. The line 
cuts at a second point 
and the perpendicular from to cuts at 
. The lines 
and 
are tangent to the circle at 
and 
respectively. Prove that the points 
and lie on the same circle.
Η λύση του 3ου προβλήματος (2η ημέρα): Φραγκάκης Νίκος
be an acute-angled triangle. Let be the inscribed circle with centre , be the circumscribed circle with centre and be the midpoint of the altitude where lies on . The circle be tangent to the side at the point . The line cuts at a second point and the perpendicular from to cuts at . The lines and are tangent to the circle at and respectively. Prove that the points and lie on the same circle.Η λύση του 3ου προβλήματος (2η ημέρα): Φραγκάκης Νίκος
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου