2007 Greece Junior Math Olympiad

Part of Greece Junior Math Olympiad

Subcontests

(4)

1

Problem 4 - Juniors

50

students in a class sent greeting cards to

25

of the others. Prove that there exist two students who greeted each other.

1

Problem 3 - Juniors

n

A =\sqrt{n^{2}+24}

B =\sqrt{n^{2}-9}

. Find all values of

n

A-B

1

Problem 2 - Juniors

n

is is an integer such that

4n+3

is divisible by

11,

find the form of

n

and the remainder of

n^{4}

upon division by

11

1

Problem 1 - Juniors

ABC

with the incentre

I,

the angle bisector

AD

meets the circumcircle of triangle

BIC

N\neq I

. a) Express the angles of

\triangle BCN

in terms of the angles of triangle

ABC

. b) Show that the circumcentre of triangle

BIC

is at the intersection of

AI

and the circumcentre of

ABC