Διασκεδαστικά Μαθηματικά : ▪ Turkey National Mathematical Olympiad 2013

Δευτέρα 22 Απριλίου 2013

▪ Turkey National Mathematical Olympiad 2013 - First Round

1 Let

be a triangle with incenter

, centroid

, and

. If

, and

, then what is

2 How many triples

are there such that

where

are prime numbers and

is a positive integer?

3 If the remainder is

when a polynomial with coefficients from the set

is divided by

, what is the least possible value of the coefficient of

in this polynomial?

4 The numbers

are written on unit squares of a

chessboard such that consequtive numbers are on unit squares sharing a common edge. At most how many prime numbers can a row have?

5 Let

be a point on side

of triangle

where

and

. The circle passing through the points

and

touches

. Let

be a point on the line which is passing through

and is perpendicular to

. If

, then what is

6 What is the

smallest positive integer which does not have

and

in its base-

representation?

7 What is the sum of real roots of the equation

8 How many kites are there such that all of its four vertices are vertices of a given regular icosagon (

-gon)?

9 Let

be a triangle with

, and

. Let

be points on sides

, respectively, such that

, and

. Let

be the reflections of the orthocenter of triangle

over the points

, respectively. What is the area of triangle

10 How many positive integers

are there such that there are exactly

positive odd integers that are less than

and relatively prime with

11 How many pairs of real numbers

are there such that

12 In the morning,

students study as

groups with two students in each group. In the afternoon, they study again as

groups with two students in each group. No matter how the groups in the morning or groups in the afternoon are established, if it is possible to find

students such that no two of them study together, what is the largest value of

13 Let

and

be points on side

of a triangle

with circumcenter

such that

is between

and

, and

. If

is the incenter of triangle

and

, then what is

14 Let

be the number of positive integers that divide the integer

. For all positive integral divisors

, what is the sum of numbers

15 No matter how

real numbers on the interval

are selected, if it is possible to find a scalene polygon such that its sides are equal to some of the numbers selected, what is the least possible value of

white and

red balls that are not identical are distributed randomly into

boxes which contain at most

balls. What is the probability that each box contains exactly

red ball?

17 Let

be an equilateral triangle with side length

and

be a point inside the triangle such that

What is the area of a triangle with side lengths

18 What is remainder when the sum

is divided by

19 What is the minimum value of

where

is a real number?

20 The numbers

are written on

stones weighing

grams such that each number is used exactly once. We have a two-pan balance that shows the difference between the weights at the left and the right pans. No matter how the numbers are written, if it is possible to determine in

weighings whether the weight of each stone is equal to the number that is written on the stone, what is the least possible value of

21 Let

and

be points on side

of a right triangle with

such that

and

. Let

be the second intersection point of the circumcircles of triangles

and

. If

, what is

22 For how many integers

, is

divisible by

23 If the conditions

hold for all real numbers

, what is

stones weighing

grams are divided into

groups such that total weights of each group are different from each other and each group contains less stones than groups with smaller total weights. For how many

, is such a division possible?

25 Let

be a point on side

of triangle

with

such that

is an angle bisector and

. Let

be a point on the extension of

after

such that

. Let

be the midpoint of

. If

is the intersection of lines

and

, what is

26 What is the maximum number of primes that divide both the numbers

and

where

is a positive integer?

27 For how many pairs

from

, the polynomial

has exactly one real root?

28 In the beginning, there is a pair of positive integers

written on the board. Alice and Bob are playing a turn-based game with the following move. At each turn, a player erases one of the numbers written on the board, and writes a different positive number not less than the half of the erased one. If a player cannot write a new number at some turn, he/she loses the game. For how many starting pairs

from the pairs

, can Alice guarantee to win when she makes the first move?

29 Let

be the circumcenter of triangle

with

. Let

be the reflections of

over the lines

, respectively. What is the distance between

and the circumcenter of triangle

30 For how many postive integers

less than

, does

divide

where

is the least prime divisor of

31 Let

be a real sequence such that

for every

. If

and

, what is

32 How many

-digit positive integers containing only the numbers

can be written such that the first and the last digits are same, and no two consecutive digits are same?

33 Let

be a point on side

of triangle

such that

is an angle bisector,

, and

. Let

be a point on side

and different than

such that

. If the perpendicular bisector of segment

meets the line

, what is

34 How many triples of positive integers

are there such that

35 What is the least positive integer

such that

where

? (

denotes the greatest integer not exceeding the real number

36 A chess club consists of at least

and at most

members, where

of them are female, and

of them are male with

. In a chess tournament, each member plays with any other member exactly one time. At each game, the winner gains

, the loser gains

and both player gains

point when a tie occurs. At the tournament, it is observed that each member gained exactly half of his/her points from the games played against male members. How many different values can

take?

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