Ημέρα 1η
1. Let 
be a triangle with circumcentre 
. The points 
lie in the interiors of the sides 
respectively, such that 
is perpendicular to 
and 
is perpendicular to 
. (By interior we mean, for example, that the point 
lies on the line 
and is between 
and 
on that line.)
be a triangle with circumcentre . The points lie in the interiors of the sides respectively, such that is perpendicular to and is perpendicular to . (By interior we mean, for example, that the point lies on the line and is between and on that line.)Let 
be the circumcentre of triangle 
. Prove that the lines 
and are perpendicular.
be the circumcentre of triangle . Prove that the lines and are perpendicular. 
2. Let 
be a positive integer. Find the greatest possible integer 
, in terms of , with the following property: a table with rows and columns can be filled with real numbers in such a manner that for any two different rows 
and 
the following holds:
be a positive integer. Find the greatest possible integer , in terms of , with the following property: a table with rows and columns can be filled with real numbers in such a manner that for any two different rows and the following holds:
4 A set 
of integers is called sum-full if 
, i.e. each element 
is the sum of some pair of (not necessarily different) elements 
. A set of integers is said to be zero-sum-free if 
is the only integer that cannot be expressed as the sum of the elements of a finite nonempty subset of .
of integers is called sum-full if , i.e. each element is the sum of some pair of (not necessarily different) elements . A set of integers is said to be zero-sum-free if is the only integer that cannot be expressed as the sum of the elements of a finite nonempty subset of .Does there exist a sum-full zero-sum-free set of integers?
for some positive integer . Find all possible values of 
.
. Find all possible values of . 2. There are infinitely many people registered on the social network Mugbook. Some pairs of (different) users are registered as friends, but each person has only finitely many friends. Every user has at least one friend. (Friendship is symmetric; that is, if is a friend of , then is a friend of .)
is a friend of , then is a friend of .)Each person is required to designate one of their friends as their best friend. If designates as her best friend, then (unfortunately) it does not follow that necessarily designates as her best friend. Someone designated as a best friend is called a 
-best friend. More generally, if 
is a positive integer, then a user is an -best friend provided that they have been designated the best friend of someone who is an 
-best friend. Someone who is a 
-best friend for every positive integer is called popular.
designates as her best friend, then (unfortunately) it does not follow that necessarily designates as her best friend. Someone designated as a best friend is called a -best friend. More generally, if is a positive integer, then a user is an -best friend provided that they have been designated the best friend of someone who is an -best friend. Someone who is a -best friend for every positive integer is called popular.(a) Prove that every popular person is the best friend of a popular person.
(b) Show that if people can have infinitely many friends, then it is possible that a popular person is not the best friend of a popular person.
3. Let be an acute-angled triangle with circumcircle 
and orthocentre 
. Let be a point of on the other side of from . Let 
be the reflection of in the line 
, and let 
be the reflection of in the line . Let 
be the second point of intersection of with the circumcircle of triangle 
.
be an acute-angled triangle with circumcircle and orthocentre . Let be a point of on the other side of from . Let be the reflection of in the line , and let be the reflection of in the line . Let be the second point of intersection of with the circumcircle of triangle .Show that the lines 
, 
and are concurrent. (The orthocentre of a triangle is the point on all three of its altitudes.)
, and are concurrent. (The orthocentre of a triangle is the point on all three of its altitudes.) 4. A word is a finite sequence of letters from some alphabet. A word is repetitive if it is a concatenation of at least two identical subwords (for example, 
and 
are repetitive, but 
and 
are not). Prove that if a word has the property that swapping any two adjacent letters makes the word repetitive, then all its letters are identical. (Note that one may swap two adjacent identical letters, leaving a word unchanged.)
and are repetitive, but and are not). Prove that if a word has the property that swapping any two adjacent letters makes the word repetitive, then all its letters are identical. (Note that one may swap two adjacent identical letters, leaving a word unchanged.)Κάντε κλικ εδώ, για να κατεβάσετε τα θέματα.
Κάντε κλικ στον παρακάτω σύνδεσμο, για να δείτε τη λύση του 1ου θέματος, της πρώτης ημέρας:
Φραγκάκης Νίκος - 2ο Λύκειο Ιεράπετρας.
Κάντε κλικ στον παρακάτω σύνδεσμο, για να δείτε τη λύση του 1ου θέματος, της πρώτης ημέρας:
Φραγκάκης Νίκος - 2ο Λύκειο Ιεράπετρας.
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