2011 Danube Mathematical Competition

Part of Danube Competition in Mathematics

Subcontests

(4)

1

max no of edges a triangle-free Hamiltonian simple graph on n vertices

Given a positive integer number

n

, determine the maximum number of edges a triangle-free Hamiltonian simple graph on

n

vertices may have.

1

sum of squares of any n prime numbers >3 is divisible by n

Determine all positive integer numbers

n

satisfying the following condition: the sum of the squares of any

n

prime numbers greater than

3

is divisible by

n

1

equal angles with an interior point of a parallelogram, easy danube geometry

ABCM

be a quadrilateral and

D

be an interior point such that

ABCD

is a parallelogram. It is known that

\angle AMB =\angle CMD

\angle MAD =\angle MCD

.

1

2- Donova Mathematical Olympiad 2011 ( Romania)

Let S be a set of positive integers such that: min { lcm (x, y) : x, y ∈ S,

x \neq y

\ge

2 + max S. Prove that

\displaystyle\sum\limits_{x \in S} \frac{1}{x} \le \frac{3}{2}