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Problems
Contests
International Contests
Middle European Mathematical Olympiad
2009 Middle European Mathematical Olympiad
2009 Middle European Mathematical Olympiad
Part of
Middle European Mathematical Olympiad
Subcontests
(12)
12
1
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2^x+2009=3^y5^z
Find all non-negative integer solutions of the equation 2^x\plus{}2009\equal{}3^y5^z.
11
1
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(m + n)^4 = m^2 n^2 + m^2 + n^2 + 6mn
Find all pairs
(
m
(m
(
m
,
n
)
n)
n
)
of integers which satisfy the equation (m \plus{} n)^4 \equal{} m^2n^2 \plus{} m^2 \plus{} n^2 \plus{} 6mn.
10
1
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DM and BL are parallel
Suppose that
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral and CD\equal{}DA. Points
E
E
E
and
F
F
F
belong to the segments
A
B
AB
A
B
and
B
C
BC
BC
respectively, and \angle ADC\equal{}2\angle EDF. Segments
D
K
DK
DK
and
D
M
DM
D
M
are height and median of triangle
D
E
F
DEF
D
EF
, respectively.
L
L
L
is the point symmetric to
K
K
K
with respect to
M
M
M
. Prove that the lines
D
M
DM
D
M
and
B
L
BL
B
L
are parallel.
9
1
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Congruent triangles
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram with \angle BAD \equal{} 60 and denote by
E
E
E
the intersection of its diagonals. The circumcircle of triangle
A
C
D
ACD
A
C
D
meets the line
B
A
BA
B
A
at
K
≠
A
K \ne A
K
=
A
, the line
B
D
BD
B
D
at
P
≠
D
P \ne D
P
=
D
and the line
B
C
BC
BC
at
L
≠
C
L\ne C
L
=
C
. The line
E
P
EP
EP
intersects the circumcircle of triangle
C
E
L
CEL
CE
L
at points
E
E
E
and
M
M
M
. Prove that triangles
K
L
M
KLM
K
L
M
and
C
A
P
CAP
C
A
P
are congruent.
8
1
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Connected colours on 2009 x 2009 board
We colour every square of the
2009
2009
2009
x
2009
2009
2009
board with one of
n
n
n
colours (we do not have to use every colour). A colour is called connected if either there is only one square of that colour or any two squares of the colour can be reached from one another by a sequence of moves of a chess queen without intermediate stops at squares having another colour (a chess quen moves horizontally, vertically or diagonally). Find the maximum
n
n
n
, such that for every colouring of the board at least on colour present at the board is connected.
7
1
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Numbers on a blackboard
The numbers
0
0
0
,
1
1
1
,
…
\dots
…
,
n
n
n
(
n
≥
2
n \ge 2
n
≥
2
) are written on a blackboard. In each step we erase an integer which is the arithmetic mean of two different numbers which are still left on the blackboard. We make such steps until no further integer can be erased. Let
g
(
n
)
g(n)
g
(
n
)
be the smallest possible number of integers left on the blackboard at the end. Find
g
(
n
)
g(n)
g
(
n
)
for every
n
n
n
.
6
1
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Determine all possible values of a^2+b^2+c^2
Let
a
a
a
,
b
b
b
,
c
c
c
be real numbers such that for every two of the equations x^2\plus{}ax\plus{}b\equal{}0, x^2\plus{}bx\plus{}c\equal{}0, x^2\plus{}cx\plus{}a\equal{}0 there is exactly one real number satisfying both of them. Determine all possible values of a^2\plus{}b^2\plus{}c^2.
5
1
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x^4+y^4+z^4+16(x^2+y^2+z^2) \ge 8(x^3+y^3+z^3)+27
Let
x
x
x
,
y
y
y
,
z
z
z
be real numbers satisfying x^2\plus{}y^2\plus{}z^2\plus{}9\equal{}4(x\plus{}y\plus{}z). Prove that x^4\plus{}y^4\plus{}z^4\plus{}16(x^2\plus{}y^2\plus{}z^2) \ge 8(x^3\plus{}y^3\plus{}z^3)\plus{}27 and determine when equality holds.
4
1
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n^{n-1}-m^{m-1} is not divisible by k
Determine all integers
k
≥
2
k\ge 2
k
≥
2
such that for all pairs
(
m
(m
(
m
,
n
)
n)
n
)
of different positive integers not greater than
k
k
k
, the number n^{n\minus{}1}\minus{}m^{m\minus{}1} is not divisible by
k
k
k
.
3
1
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\angle AGH = \angle DHG in convex quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral such that
A
B
AB
A
B
and
C
D
CD
C
D
are not parallel and AB\equal{}CD. The midpoints of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
are
E
E
E
and
F
F
F
, respectively. The line
E
F
EF
EF
meets segments
A
B
AB
A
B
and
C
D
CD
C
D
at
G
G
G
and
H
H
H
, respectively. Show that \angle AGH \equal{} \angle DHG.
2
1
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Colour the edges of a polygon
Suppose that we have
n
≥
3
n \ge 3
n
≥
3
distinct colours. Let
f
(
n
)
f(n)
f
(
n
)
be the greatest integer with the property that every side and every diagonal of a convex polygon with
f
(
n
)
f(n)
f
(
n
)
vertices can be coloured with one of
n
n
n
colours in the following way: (i) At least two colours are used, (ii) any three vertices of the polygon determine either three segments of the same colour or of three different colours. Show that f(n) \le (n\minus{}1)^2 with equality for infintely many values of
n
n
n
.
1
1
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f(xf(y)) + f(f(x) + f(y)) = yf(x) + f(x + f(y))
Find all functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
, such that f(xf(y)) \plus{} f(f(x) \plus{} f(y)) \equal{} yf(x) \plus{} f(x \plus{} f(y)) holds for all
x
x
x
,
y
∈
R
y \in \mathbb{R}
y
∈
R
, where
R
\mathbb{R}
R
denotes the set of real numbers.