1996 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

In a convex pentagon ABCDE, 4 segments are concurrent

In a convex pentagon

ABCDE

M

N

P

Q

R

are the midpoints of the sides

AB

BC

CD

DE

EA

, respectively. If the segments

AP

BQ

CR

DM

pass through a single point, prove that

EN

contains that point as well. Yugoslavia

1

Geometrical inequality involving the centroid and circumcent

O

be the circumcenter and

G

be the centroid of a triangle

ABC

R

r

are the circumcenter and incenter of the triangle, respectively, prove that

OG \leq \sqrt{ R ( R - 2r ) } .

1

Beautiful subset

X=\{1,2, \ldots, 2^{1996}-1\}

, prove that there exist a subset

A

that satisfies these conditions: a)

1\in A

2^{1996}-1\in A

; b) Every element of

A

1

is equal to the sum of two (possibly equal) elements from

A

; c) The maximum number of elements of

A

2012

1

p>5 prime, then p-x^2 | p-y^2

p

be a prime number with

p>5

. Consider the set X \equal{} \left\{p \minus{} n^2 \mid n\in \mathbb{N} ,\ n^2 < p\right\}. Prove that the set

X

has two distinct elements

x

y

x\neq 1

x\mid y