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Problems
Contests
International Contests
Middle European Mathematical Olympiad
2008 Middle European Mathematical Olympiad
2008 Middle European Mathematical Olympiad
Part of
Middle European Mathematical Olympiad
Subcontests
(4)
4
2
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4n+1 and kn+1 have no common divisor
Determine that all
k
∈
Z
k \in \mathbb{Z}
k
∈
Z
such that
∀
n
\forall n
∀
n
the numbers 4n\plus{}1 and kn\plus{}1 have no common divisor.
Sum of all positive divisors of n is a power of two
Prove: If the sum of all positive divisors of n \in \mathbb{Z}^{\plus{}} is a power of two, then the number/amount of the divisors is a power of two.
3
2
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Prove that DK = DL
Let
A
B
C
ABC
A
BC
be an isosceles triangle with AC \equal{} BC. Its incircle touches
A
B
AB
A
B
in
D
D
D
and
B
C
BC
BC
in
E
.
E.
E
.
A line distinct of
A
E
AE
A
E
goes through
A
A
A
and intersects the incircle in
F
F
F
and
G
.
G.
G
.
Line
A
B
AB
A
B
intersects line
E
F
EF
EF
and
E
G
EG
EG
in
K
K
K
and
L
,
L,
L
,
respectively. Prove that DK \equal{} DL.
D is an interior point of segment AE
Let
A
B
C
ABC
A
BC
be an acute-angled triangle. Let
E
E
E
be a point such
E
E
E
and
B
B
B
are on distinct sides of the line
A
C
,
AC,
A
C
,
and
D
D
D
is an interior point of segment
A
E
.
AE.
A
E
.
We have \angle ADB \equal{} \angle CDE, \angle BAD \equal{} \angle ECD, and \angle ACB \equal{} \angle EBA. Prove that
B
,
C
B, C
B
,
C
and
E
E
E
lie on the same line.
2
2
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Put 2n - 2 identical pebbles on the checkerboard
Consider a
n
×
n
n \times n
n
×
n
checkerboard with
n
>
1
,
n
∈
N
.
n > 1, n \in \mathbb{N}.
n
>
1
,
n
∈
N
.
How many possibilities are there to put 2n \minus{} 2 identical pebbles on the checkerboard (each on a different field/place) such that no two pebbles are on the same checkerboard diagonal. Two pebbles are on the same checkerboard diagonal if the connection segment of the midpoints of the respective fields are parallel to one of the diagonals of the
n
×
n
n \times n
n
×
n
square.
Replace both of them by their sum
On a blackboard there are n \geq 2, n \in \mathbb{Z}^{\plus{}} numbers. In each step we select two numbers from the blackboard and replace both of them by their sum. Determine all numbers
n
n
n
for which it is possible to yield
n
n
n
identical number after a finite number of steps.
1
2
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Determine the least possible value of a_(2008)
Let (a_n)^{\infty}_{n\equal{}1} be a sequence of integers with a_{n} < a_{n\plus{}1}, \forall n \geq 1. For all quadruple
(
i
,
j
,
k
,
l
)
(i,j,k,l)
(
i
,
j
,
k
,
l
)
of indices such that
1
≤
i
<
j
≤
k
<
l
1 \leq i < j \leq k < l
1
≤
i
<
j
≤
k
<
l
and i \plus{} l \equal{} j \plus{} k we have the inequality a_{i} \plus{} a_{l} > a_{j} \plus{} a_{k}. Determine the least possible value of
a
2008
.
a_{2008}.
a
2008
.
xf(x + xy) = xf(x) + f(x^2)f(y)
Determine all functions
f
:
R
↦
R
f: \mathbb{R} \mapsto \mathbb{R}
f
:
R
↦
R
such that x f(x \plus{} xy) \equal{} x f(x) \plus{} f \left( x^2 \right) f(y) \forall x,y \in \mathbb{R}.