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International Contests
Middle European Mathematical Olympiad
2010 Middle European Mathematical Olympiad
2010 Middle European Mathematical Olympiad
Part of
Middle European Mathematical Olympiad
Subcontests
(12)
12
1
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MEMO 2010, Problem T-8: Existece of m
We are given a positive integer
n
n
n
which is not a power of two. Show that ther exists a positive integer
m
m
m
with the following two properties: (a)
m
m
m
is the product of two consecutive positive integers; (b) the decimal representation of
m
m
m
consists of two identical blocks with
n
n
n
digits.(4th Middle European Mathematical Olympiad, Team Competition, Problem 8)
11
1
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MEMO 2010, Problem T-7: Decomposition into cubes and squares
For a nonnegative integer
n
n
n
, define
a
n
a_n
a
n
to be the positive integer with decimal representation 1\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}1\mbox{.} Prove that
a
n
3
\frac{a_n}{3}
3
a
n
is always the sum of two positive perfect cubes but never the sum of two perfect squares.(4th Middle European Mathematical Olympiad, Team Competition, Problem 7)
10
1
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MEMO 2010, Problem T-6: Equal angles
Let
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
,
E
E
E
be points such that
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral and
A
B
D
E
ABDE
A
B
D
E
is a parallelogram. The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at
S
S
S
and the rays
A
B
AB
A
B
and
D
C
DC
D
C
intersect at
F
F
F
. Prove that
∢
A
F
S
=
∢
E
C
D
\sphericalangle{AFS}=\sphericalangle{ECD}
∢
A
FS
=
∢
EC
D
.(4th Middle European Mathematical Olympiad, Team Competition, Problem 6)
9
1
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MEMO 2010, Problem T-5: Parallel lines
The incircle of the triangle
A
B
C
ABC
A
BC
touches the sides
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
in the points
D
D
D
,
E
E
E
and
F
F
F
, respectively. Let
K
K
K
be the point symmetric to
D
D
D
with respect to the incenter. The lines
D
E
DE
D
E
and
F
K
FK
F
K
intersect at
S
S
S
. Prove that
A
S
AS
A
S
is parallel to
B
C
BC
BC
.(4th Middle European Mathematical Olympiad, Team Competition, Problem 5)
8
1
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MEMO 2010, Problem T-4: Coloring vertices
Let
n
n
n
be a positive integer. A square
A
B
C
D
ABCD
A
BC
D
is partitioned into
n
2
n^2
n
2
unit squares. Each of them is divided into two triangles by the diagonal parallel to
B
D
BD
B
D
. Some of the vertices of the unit squares are colored red in such a way that each of these
2
n
2
2n^2
2
n
2
triangles contains at least one red vertex. Find the least number of red vertices.(4th Middle European Mathematical Olympiad, Team Competition, Problem 4)
7
1
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MEMO 2010, Problem T-3: Shooting Fortresses & Coprimality
In each vertex of a regular
n
n
n
-gon, there is a fortress. At the same moment, each fortress shoots one of the two nearest fortresses and hits it. The result of the shooting is the set of the hit fortresses; we do not distinguish whether a fortress was hit once or twice. Let
P
(
n
)
P(n)
P
(
n
)
be the number of possible results of the shooting. Prove that for every positive integer
k
⩾
3
k\geqslant 3
k
⩾
3
,
P
(
k
)
P(k)
P
(
k
)
and
P
(
k
+
1
)
P(k+1)
P
(
k
+
1
)
are relatively prime.(4th Middle European Mathematical Olympiad, Team Competition, Problem 3)
6
1
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MEMO 2010, Problem T-2: Stronger than Cauchy
For each integer
n
⩾
2
n\geqslant2
n
⩾
2
, determine the largest real constant
C
n
C_n
C
n
such that for all positive real numbers
a
1
,
…
,
a
n
a_1, \ldots, a_n
a
1
,
…
,
a
n
we have \frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}(4th Middle European Mathematical Olympiad, Team Competition, Problem 2)
5
1
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MEMO 2010, Problem T-1: Increasing sequences
Three strictly increasing sequences
a
1
,
a
2
,
a
3
,
…
,
b
1
,
b
2
,
b
3
,
…
,
c
1
,
c
2
,
c
3
,
…
a_1, a_2, a_3, \ldots,\qquad b_1, b_2, b_3, \ldots,\qquad c_1, c_2, c_3, \ldots
a
1
,
a
2
,
a
3
,
…
,
b
1
,
b
2
,
b
3
,
…
,
c
1
,
c
2
,
c
3
,
…
of positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer
n
n
n
, the following conditions hold: (a)
c
a
n
=
b
n
+
1
c_{a_n}=b_n+1
c
a
n
=
b
n
+
1
; (b)
a
n
+
1
>
b
n
a_{n+1}>b_n
a
n
+
1
>
b
n
; (c) the number
c
n
+
1
c
n
−
(
n
+
1
)
c
n
+
1
−
n
c
n
c_{n+1}c_{n}-(n+1)c_{n+1}-nc_n
c
n
+
1
c
n
−
(
n
+
1
)
c
n
+
1
−
n
c
n
is even. Find
a
2010
a_{2010}
a
2010
,
b
2010
b_{2010}
b
2010
and
c
2010
c_{2010}
c
2010
.(4th Middle European Mathematical Olympiad, Team Competition, Problem 1)
4
1
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MEMO 2010, Problem I-4: Difference of divisors divides n
Find all positive integers
n
n
n
which satisfy the following tow conditions: (a)
n
n
n
has at least four different positive divisors; (b) for any divisors
a
a
a
and
b
b
b
of
n
n
n
satisfying
1
<
a
<
b
<
n
1<a<b<n
1
<
a
<
b
<
n
, the number
b
−
a
b-a
b
−
a
divides
n
n
n
.(4th Middle European Mathematical Olympiad, Individual Competition, Problem 4)
3
1
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MEMO 2010, Problem I-3: Three lines meet at one point
We are given a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
with a point
E
E
E
on the diagonal
A
C
AC
A
C
such that
A
D
=
A
E
AD=AE
A
D
=
A
E
and
C
B
=
C
E
CB=CE
CB
=
CE
. Let
M
M
M
be the center of the circumcircle
k
k
k
of the triangle
B
D
E
BDE
B
D
E
. The circle
k
k
k
intersects the line
A
C
AC
A
C
in the points
E
E
E
and
F
F
F
. Prove that the lines
F
M
FM
FM
,
A
D
AD
A
D
and
B
C
BC
BC
meet at one point. (4th Middle European Mathematical Olympiad, Individual Competition, Problem 3)
2
1
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MEMO 2010, Problem I-2: A game
All positive divisors of a positive integer
N
N
N
are written on a blackboard. Two players
A
A
A
and
B
B
B
play the following game taking alternate moves. In the firt move, the player
A
A
A
erases
N
N
N
. If the last erased number is
d
d
d
, then the next player erases either a divisor of
d
d
d
or a multiple of
d
d
d
. The player who cannot make a move loses. Determine all numbers
N
N
N
for which
A
A
A
can win independently of the moves of
B
B
B
.(4th Middle European Mathematical Olympiad, Individual Competition, Problem 2)
1
1
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Find all functions f with f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y)
Find all functions
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that for all
x
,
y
∈
R
x, y\in\mathbb{R}
x
,
y
∈
R
, we have
f
(
x
+
y
)
+
f
(
x
)
f
(
y
)
=
f
(
x
y
)
+
(
y
+
1
)
f
(
x
)
+
(
x
+
1
)
f
(
y
)
.
f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).
f
(
x
+
y
)
+
f
(
x
)
f
(
y
)
=
f
(
x
y
)
+
(
y
+
1
)
f
(
x
)
+
(
x
+
1
)
f
(
y
)
.