MathDB

Higher Secondary: 2011
Time: 4 Hours
Problem 1: Prove that for any non-negative integer

n

the numbers

1, 2, 3, ..., 4n

can be divided in tow mutually exclusive classes with equal number of members so that the sum of numbers of each class is equal. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=709
Problem 2: In the first round of a chess tournament, each player plays against every other player exactly once. A player gets

3, 1

-1

points respectively for winning, drawing or losing a match. After the end of the first round, it is found that the sum of the scores of all the players is

90

. How many players were there in the tournament? http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=708
Problem 3:

E

is the midpoint of side

BC

of rectangle

ABCD

A

point

X

is chosen on

BE

DX

meets extended

AB

P

. Find the position of

X

so that the sum of the areas of

\triangle BPX

and

\triangle DXC

is maximum with proof. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=683
Problem 4: Which one is larger 2011! or,

(1006)^{2011}

? Justify your answer. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=707
Problem 5: In a scalene triangle

ABC

with

\angle A = 90^{\circ}

, the tangent line at

A

to its circumcircle meets line

BC

M

and the incircle touches

AC

S

and

AB

R

. The lines

RS

and

BC

intersect at

N

while the lines

AM

and

SR

intersect at

U

. Prove that the triangle

UMN

is isosceles. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=706
Problem 6:

p

is a prime and sum of the numbers from

1

p

is divisible by all primes less or equal to

p

. Find the value of

p

with proof. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=693
Problem 7: Consider a group of

n > 1

people. Any two people of this group are related by mutual friendship or mutual enmity. Any friend of a friend and any enemy of an enemy is a friend. If

A

and

B

are friends/enemies then we count it as

1

friendship/enmity. It is observed that the number of friendships and number of enmities are equal in the group. Find all possible values of

n

. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=694
Problem 8:

ABC

is a right angled triangle with

\angle A = 90^{\circ}

and

D

be the midpoint of

BC

. A point

F

is chosen on

AB

CA

and

DF

meet at

G

and

GB \parallel AD

CF

and

AD

meet at

O

and

AF = FO

GO

meets

BC

R

. Find the sides of

ABC

if the area of

GDR

\dfrac{2}{\sqrt{15}}

http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=704
Problem 9: The repeat of a natural number is obtained by writing it twice in a row (for example, the repeat of

123

123123

). Find a positive integer (if any) whose repeat is a perfect square. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=703
Problem 10: Consider a square grid with

n

rows and

n

columns, where

n

is odd (similar to a chessboard). Among the

n^2

squares of the grid,

p

are black and the others are white. The number of black squares is maximized while their arrangement is such that horizontally, vertically or diagonally neighboring black squares are separated by at least one white square between them. Show that there are infinitely many triplets of integers

(p, q, n)

so that the number of white squares is

q^2

. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=702
The problems of the Junior categories are available in [url=http://matholympiad.org.bd/forum/]BdMO Online forum: http://matholympiad.org.bd/forum/viewtopic.php?f=25&t=678

HS

Problems(1)