1. A sequence 
of natural numbers is defined by the rule
of natural numbers is defined by the rule
where 
is the last digit of 
. Prove that such a sequence contains infinitely many powers of 
if and only if 
is not divisible by 
.
is the last digit of . Prove that such a sequence contains infinitely many powers of if and only if is not divisible by . 2. Find all quadruples 
of positive real numbers such that
of positive real numbers such that
and 
. 
3. In triangle 
the midpoint of 
is called 
. Let 
be a variable interior point of the triangle such that 
. Let 
be the circumcircle of triangle 
. The line 
intersects a second time at 
. Define 
as the reflection of in the tangent to at 
. Prove that the length 
is independent of the position of inside the triangle. 4. Yesterday, 
people sat around a round table. Each participant remembers only who his two neighbours were, but not necessarily which one sat on his left and which one sat on his right.
Today, you would like the same people to sit around the same round table so that each participant has the same two neighbours as yesterday (it is possible that yesterday’s left-hand side neighbour is today’s right-hand side neighbour). You are allowed to query some of the participants: if anyone is asked, he will answer by pointing at his two neighbours from yesterday.
people sat around a round table. Each participant remembers only who his two neighbours were, but not necessarily which one sat on his left and which one sat on his right.Today, you would like the same people to sit around the same round table so that each participant has the same two neighbours as yesterday (it is possible that yesterday’s left-hand side neighbour is today’s right-hand side neighbour). You are allowed to query some of the participants: if anyone is asked, he will answer by pointing at his two neighbours from yesterday.
a) Determine the minimal number 
of participants you have to query in order to be certain to succeed, if later questions must not depend on the outcome of the previous questions. That is, you have to choose in advance the list of people you are going to query, before effectively asking any question.
of participants you have to query in order to be certain to succeed, if later questions must not depend on the outcome of the previous questions. That is, you have to choose in advance the list of people you are going to query, before effectively asking any question.b) Determine the minimal number 
of participants you have to query in order to be certain to succeed, if later questions may depend on the outcome of previous questions. That is, you can wait until you get the first answer to choose whom to ask the second question, and so on.
of participants you have to query in order to be certain to succeed, if later questions may depend on the outcome of previous questions. That is, you can wait until you get the first answer to choose whom to ask the second question, and so on.Κάντε κλικ εδώ, για να κατεβάσετε τα θέματα.
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου