MathDB
Problems
Contests
International Contests
Danube Competition in Mathematics
2011 Danube Mathematical Competition
2011 Danube Mathematical Competition
Part of
Danube Competition in Mathematics
Subcontests
(4)
4
1
Hide problems
max no of edges a triangle-free Hamiltonian simple graph on n vertices
Given a positive integer number
n
n
n
, determine the maximum number of edges a triangle-free Hamiltonian simple graph on
n
n
n
vertices may have.
3
1
Hide problems
sum of squares of any n prime numbers >3 is divisible by n
Determine all positive integer numbers
n
n
n
satisfying the following condition: the sum of the squares of any
n
n
n
prime numbers greater than
3
3
3
is divisible by
n
n
n
.
1
1
Hide problems
equal angles with an interior point of a parallelogram, easy danube geometry
Let
A
B
C
M
ABCM
A
BCM
be a quadrilateral and
D
D
D
be an interior point such that
A
B
C
D
ABCD
A
BC
D
is a parallelogram. It is known that
∠
A
M
B
=
∠
C
M
D
\angle AMB =\angle CMD
∠
A
MB
=
∠
CM
D
. Prove that
∠
M
A
D
=
∠
M
C
D
\angle MAD =\angle MCD
∠
M
A
D
=
∠
MC
D
.
2
1
Hide problems
2- Donova Mathematical Olympiad 2011 ( Romania)
Let S be a set of positive integers such that: min { lcm (x, y) : x, y ∈ S,
x
≠
y
x \neq y
x
=
y
}
≥
\ge
≥
2 + max S. Prove that
∑
x
∈
S
1
x
≤
3
2
\displaystyle\sum\limits_{x \in S} \frac{1}{x} \le \frac{3}{2}
x
∈
S
∑
x
1
≤
2
3
.