1.Find the perimeter of a triangle whose altitudes are 
and 
.
is
?
and .
3. Which one satisfies the equation 
?
?
5. 
is given with 
, and 
. Let 
be a point on 
such that 
. Let 
and 
be the inradii of 
and 
, respectively. What is 
?
is given with , and . Let be a point on such that . Let and be the inradii of and , respectively. What is ?
6. Which one statisfies 
?
?
8. In how many different ways can one select two distinct subsets of the set 
, so that one includes the other?
, so that one includes the other?
9. The chord 
of the circle with diameter 
is perpendicular to . Let 
and 
be the midpoints of and 
, respectively. If 
and 
, then 
of the circle with diameter is perpendicular to . Let and be the midpoints of and , respectively. If and , then
10. How many positive integers 
are there such that there are 
positive integers that are less than and relatively prime with ?
are there such that there are positive integers that are less than and relatively prime with ?
11. The number of real quadruples 
satisfying
satisfying is
12. How many subsets of the set 
are there that does not contain 4 consequtive integers?
are there that does not contain 4 consequtive integers?
13. points with no three collinear are given. How many obtuse triangles can be formed by these points?
points with no three collinear are given. How many obtuse triangles can be formed by these points?
a) b) c) 
d) 
e) 
b) c) d) e) 
16. Every cell of 
chessboard contains either 
or 
. It is known that there are at least four rows such that the sum of numbers inside the cells of those rows is positive. At most how many columns are there such that the sum of numbers inside the cells of those columns is less than 
?
chessboard contains either or . It is known that there are at least four rows such that the sum of numbers inside the cells of those rows is positive. At most how many columns are there such that the sum of numbers inside the cells of those columns is less than ?
18. If the representation of a positive number as a product of powers of distinct prime numbers contains no even powers other than 
s, we will call the number singular. At most how many consequtive singular numbers are there?
s, we will call the number singular. At most how many consequtive singular numbers are there?
19. What is the sum of real roots of the equation 
?
?
20. For each permutation 
of the numbers 
, we can determine at least 
of 
s when we get 
. can be at most ?
of the numbers , we can determine at least of s when we get . can be at most ?
21. The angle bisector of vertex 
of cuts at . The circle passing through and touching to
at meets and 
at 
and
, respectively. 
and 
meet at 
.
If
, then 
of cuts at . The circle passing through and touching to at meets and at and, respectively. and meet at .If
, then 
24. There are 
backgammon checkers (stones, pieces) with one side is black and the other side is white.
backgammon checkers (stones, pieces) with one side is black and the other side is white.
These checkers are arranged into a line such that no two consequtive checkers are in same color. At each move, we are chosing two checkers. And we are turning upside down of the two checkers and all of the checkers between the two. At least how many moves are required to make all checkers same color?
26. How many prime numbers less than 
can be represented as sum of squares of consequtive positive integers?
can be represented as sum of squares of consequtive positive integers?
28. At the beginning, three boxes contains 
, , and pieces, respectively. Ayşe and Burak are playing a turn-based game with these pieces. At each turn, the player takes at least one piece from one of the boxes. The player who takes the last piece will win the game. Ayşe will be the first player. They are playing the game once for each 
, 
, 
, 
, 
. In how many of them can Ayşe guarantee to win the game?
, , and pieces, respectively. Ayşe and Burak are playing a turn-based game with these pieces. At each turn, the player takes at least one piece from one of the boxes. The player who takes the last piece will win the game. Ayşe will be the first player. They are playing the game once for each , , , , . In how many of them can Ayşe guarantee to win the game?
If 
, then 
, then 
?
33. Let 
be a rectangular prism with 
. 
is a point on the edge 
satisfying 
. Let 
and 
be the foot of the altitudes from at 
and 
, respectively. If 
, then 
be a rectangular prism with . is a point on the edge satisfying . Let and be the foot of the altitudes from at and , respectively. If , then 
35. For every positive real pair 
satisfying the equation 
, if the greatest value of 
is , and the greatest value of 
is 
, then 
satisfying the equation , if the greatest value of is , and the greatest value of is , then
36. stones are put into boxes in such a way that each box has at most stones. We are chosing some of the boxes. We are throwing some of the stones of the chosen boxes. Whatever the first arrangement of the stones inside the boxes is, if we can guarantee that there are equal stones inside the chosen boxes and the sum of them is at least , then can be at least?
stones are put into boxes in such a way that each box has at most stones. We are chosing some of the boxes. We are throwing some of the stones of the chosen boxes. Whatever the first arrangement of the stones inside the boxes is, if we can guarantee that there are equal stones inside the chosen boxes and the sum of them is at least , then can be at least?
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