TST 1
Hμέρα 1η
, 
and 
. Prove that
. 
2. Given a scalene triangle 
. Its incircle touches 
at 
respectvely. Let 
be the symmetric points of 
with 
,of 
with 
,of 
with 
,respectively. Line 
intersects 
at 
,line 
intersects 
at 
,line 
intersects 
at 
. Prove that 
are collinear.
. Its incircle touches at respectvely. Let be the symmetric points of with ,of with ,of with ,respectively. Line intersects at ,line intersects at ,line intersects at . Prove that are collinear. 3. Let 
for all 
. Prove there exist infinitely many finite sets 
of positive integers, satisfying 
, and
for all . Prove there exist infinitely many finite sets of positive integers, satisfying , and
Hμέρα 2η
1. Given two circles 
, 
denotes all 
satisfies that 
is the circumcircle of , 
is the 
- excircle of , touches 
at .
, denotes all satisfies that is the circumcircle of , is the - excircle of , touches at . is not empty, prove that the centroid of 
is a fixed point.2. For a positive integer
, denote by 
the number of its positive divisors. For a positive integer , if 
for all 
, we call a good number. Prove that for any positive integer 
, there are only finitely many good numbers not divisible by . 
TST 2
Hμέρα 1η
1. In a simple graph 
, we call 
pairwise adjacent vertices a -clique. If a vertex is connected with all other vertices in the graph, we call it a centralvertex. Given are two integers 
such that 
. Let be a graph on vertices such that
, we call pairwise adjacent vertices a -clique. If a vertex is connected with all other vertices in the graph, we call it a centralvertex. Given are two integers such that . Let be a graph on vertices such that(1) does not contain a 
-clique;
does not contain a -clique;(2) if we add an arbitrary edge to , that creates a -clique.
, that creates a -clique.Find the least possible number of central vertices in .
. 2. Prove that there exists a positive real number 
with the following property: for any integer 
and any subset 
of the set 
such that 
, there exist 
(not necessarily distinct) such that
with the following property: for any integer and any subset of the set such that , there exist (not necessarily distinct) such that
where 
.
. 3. Let 
be two given integers. For any integer 
, let 
be the smallest integer which is larger than 
and can be uniquely represented as 
, where 
. Given that there are only a finite number of even numbers in 
, prove that the sequence 
is eventually periodic, i.e. that there exist positive integers 
such that for all integers 
, we have
be two given integers. For any integer , let be the smallest integer which is larger than and can be uniquely represented as , where . Given that there are only a finite number of even numbers in , prove that the sequence is eventually periodic, i.e. that there exist positive integers such that for all integers , we have
Hμέρα 2η
1. Given an integer . Prove that there only exist a finite number of n-tuples of positive integers 
which simultaneously satisfy the following three conditions:
. Prove that there only exist a finite number of n-tuples of positive integers which simultaneously satisfy the following three conditions:
; 
,where 
. 2. Given two integers 
which are greater than 
. 
are two given positive real numbers such that 
. For all 
which are not all zeroes,find the maximal value of the expression
which are greater than . are two given positive real numbers such that . For all which are not all zeroes,find the maximal value of the expression
3. Given an integer , a function 
is called good, if for any integer 
there exists an integer 
such that for every integer 
we have
, a function is called good, if for any integer there exists an integer such that for every integer we have 
Find the number of good functions.
TST 3
Hμέρα 1η
1. In an acute-angled , 
, 
is its orthocenter. 
are two points on 
respectively, such that 
. Let 
be the circumcenter of triangle 
. is a point on the same side with of such that 
is an equilateral triangle. Prove that 
are collinear.
, , is its orthocenter. are two points on respectively, such that . Let be the circumcenter of triangle . is a point on the same side with of such that is an equilateral triangle. Prove that are collinear. 2. Given an integer 
. Prove that there exist pairwise distinct positive integers 
such that for any non-negative integers 
satisfying 
and 
, we have
. Prove that there exist pairwise distinct positive integers such that for any non-negative integers satisfying and , we have
we can obtain a new polynomial 
by multiplying some of its coefficients by 
such that every root 
of satisfies the inequality
by multiplying some of its coefficients by such that every root of satisfies the inequality
Hμέρα 2η
1. Given an integer 
. 
. are two subsets of such that for every pair of 
is a perfect square. Prove that
. . are two subsets of such that for every pair of is a perfect square. Prove that
3. In some squares of a 
grid,there are some beatles such that no square contain more than one beatle. At one moment, all the beatles fly off the grid and then land on the grid again,also satisfying the condition that there is at most one beatle standing in each square. The vector from the centre of the square from which a beatle 
fly to the centreof the square on which it land is called the translation vector of beatle . For all possible starting and ending configurations, find the maximal length of the sum of the translation vectors of all beatles.
grid,there are some beatles such that no square contain more than one beatle. At one moment, all the beatles fly off the grid and then land on the grid again,also satisfying the condition that there is at most one beatle standing in each square. The vector from the centre of the square from which a beatle fly to the centreof the square on which it land is called the translation vector of beatle . For all possible starting and ending configurations, find the maximal length of the sum of the translation vectors of all beatles.
