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International Contests
Middle European Mathematical Olympiad
2012 Middle European Mathematical Olympiad
2012 Middle European Mathematical Olympiad
Part of
Middle European Mathematical Olympiad
Subcontests
(8)
8
1
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The number of positive divisors
For any positive integer
n
n
n
let
d
(
n
)
d(n)
d
(
n
)
denote the number of positive divisors of
n
n
n
. Do there exist positive integers
a
a
a
and
b
b
b
, such that
d
(
a
)
=
d
(
b
)
d(a)=d(b)
d
(
a
)
=
d
(
b
)
and
d
(
a
2
)
=
d
(
b
2
)
d(a^2 ) = d(b^2 )
d
(
a
2
)
=
d
(
b
2
)
, but
d
(
a
3
)
≠
d
(
b
3
)
d(a^3 ) \ne d(b^3 )
d
(
a
3
)
=
d
(
b
3
)
?
7
1
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Diophantine Equation
Find all triplets
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of positive integers such that
x
y
+
y
x
=
z
y
x^y + y^x = z^y
x
y
+
y
x
=
z
y
x
y
+
2012
=
y
z
+
1
x^y + 2012 = y^{z+1}
x
y
+
2012
=
y
z
+
1
6
1
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Middle European Mathematical Olympiad 2012 - Team Compt. T-6
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with no pair of parallel sides, such that
∠
A
B
C
=
∠
C
D
A
\angle ABC = \angle CDA
∠
A
BC
=
∠
C
D
A
. Assume that the intersections of the pairs of neighbouring angle bisectors of
A
B
C
D
ABCD
A
BC
D
form a convex quadrilateral
E
F
G
H
EFGH
EFG
H
. Let
K
K
K
be the intersection of the diagonals of
E
F
G
H
EFGH
EFG
H
. Prove that the lines
A
B
AB
A
B
and
C
D
CD
C
D
intersect on the circumcircle of the triangle
B
K
D
BKD
B
KD
.
5
1
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Middle European Mathematical Olympiad 2012 - Team Compt. T-5
Let
K
K
K
be the midpoint of the side
A
B
AB
A
B
of a given triangle
A
B
C
ABC
A
BC
. Let
L
L
L
and
M
M
M
be points on the sides
A
C
AC
A
C
and
B
C
BC
BC
, respectively, such that
∠
C
L
K
=
∠
K
M
C
\angle CLK = \angle KMC
∠
C
L
K
=
∠
K
MC
. Prove that the perpendiculars to the sides
A
B
,
A
C
,
AB, AC,
A
B
,
A
C
,
and
B
C
BC
BC
passing through
K
,
L
,
K,L,
K
,
L
,
and
M
M
M
, respectively, are concurrent.
4
2
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Sequence and prime divisors
The sequence
{
a
n
}
n
≥
0
\{ a_n \} _ { n \ge 0 }
{
a
n
}
n
≥
0
is defined by
a
0
=
2
,
a
1
=
4
a_0 = 2 , a_1 = 4
a
0
=
2
,
a
1
=
4
and
a
n
+
1
=
a
n
a
n
−
1
2
+
a
n
+
a
n
−
1
a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1}
a
n
+
1
=
2
a
n
a
n
−
1
+
a
n
+
a
n
−
1
for all positive integers
n
n
n
. Determine all prime numbers
p
p
p
for which there exists a positive integer
m
m
m
such that
p
p
p
divides the number
a
m
−
1
a_m - 1
a
m
−
1
.
Average value of function taken over all permutations
Let
p
>
2
p>2
p
>
2
be a prime number. For any permutation
π
=
(
π
(
1
)
,
π
(
2
)
,
⋯
,
π
(
p
)
)
\pi = ( \pi(1) , \pi(2) , \cdots , \pi(p) )
π
=
(
π
(
1
)
,
π
(
2
)
,
⋯
,
π
(
p
))
of the set
S
=
{
1
,
2
,
⋯
,
p
}
S = \{ 1, 2, \cdots , p \}
S
=
{
1
,
2
,
⋯
,
p
}
, let
f
(
π
)
f( \pi )
f
(
π
)
denote the number of multiples of
p
p
p
among the following
p
p
p
numbers:
π
(
1
)
,
π
(
1
)
+
π
(
2
)
,
⋯
,
π
(
1
)
+
π
(
2
)
+
⋯
+
π
(
p
)
\pi(1) , \pi(1) + \pi(2) , \cdots , \pi(1) + \pi(2) + \cdots + \pi(p)
π
(
1
)
,
π
(
1
)
+
π
(
2
)
,
⋯
,
π
(
1
)
+
π
(
2
)
+
⋯
+
π
(
p
)
Determine the average value of
f
(
π
)
f( \pi)
f
(
π
)
taken over all permutations
π
\pi
π
of
S
S
S
.
3
2
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Middle European Mathematical Olympiad 2012 - Individuals I-3
In a given trapezium
A
B
C
D
ABCD
A
BC
D
with
A
B
AB
A
B
parallel to
C
D
CD
C
D
and
A
B
>
C
D
AB > CD
A
B
>
C
D
, the line
B
D
BD
B
D
bisects the angle
∠
A
D
C
\angle ADC
∠
A
D
C
. The line through
C
C
C
parallel to
A
D
AD
A
D
meets the segments
B
D
BD
B
D
and
A
B
AB
A
B
in
E
E
E
and
F
F
F
, respectively. Let
O
O
O
be the circumcenter of the triangle
B
E
F
BEF
BEF
. Suppose that
∠
A
C
O
=
6
0
∘
\angle ACO = 60^{\circ}
∠
A
CO
=
6
0
∘
. Prove the equality
C
F
=
A
F
+
F
O
.
CF = AF + FO .
CF
=
A
F
+
FO
.
Considering the number of words with parity
Let
n
n
n
be a positive integer. Consider words of length
n
n
n
composed of letters from the set
{
M
,
E
,
O
}
\{ M, E, O \}
{
M
,
E
,
O
}
. Let
a
a
a
be the number of such words containing an even number (possibly 0) of blocks
M
E
ME
ME
and an even number (possibly 0) blocks of
M
O
MO
MO
. Similarly let
b
b
b
the number of such words containing an odd number of blocks
M
E
ME
ME
and an odd number of blocks
M
O
MO
MO
. Prove that
a
>
b
a>b
a
>
b
.
2
2
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3 variables inequality with abc=1
Let
a
,
b
a,b
a
,
b
and
c
c
c
be positive real numbers with
a
b
c
=
1
abc = 1
ab
c
=
1
. Prove that
9
+
16
a
2
+
9
+
16
b
2
+
9
+
16
c
2
≥
3
+
4
(
a
+
b
+
c
)
\sqrt{ 9 + 16a^2}+\sqrt{ 9 + 16b^2}+\sqrt{ 9 + 16c^2} \ge 3 +4(a+b+c)
9
+
16
a
2
+
9
+
16
b
2
+
9
+
16
c
2
≥
3
+
4
(
a
+
b
+
c
)
The maximum number of 'allowed' set
Let
N
N
N
be a positive integer. A set
S
⊂
{
1
,
2
,
⋯
,
N
}
S \subset \{ 1, 2, \cdots, N \}
S
⊂
{
1
,
2
,
⋯
,
N
}
is called allowed if it does not contain three distinct elements
a
,
b
,
c
a, b, c
a
,
b
,
c
such that
a
a
a
divides
b
b
b
and
b
b
b
divides
c
c
c
. Determine the largest possible number of elements in an allowed set
S
S
S
.
1
2
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Functional equation on positive reals
Let
R
+
\mathbb{R} ^{+}
R
+
denote the set of all positive real numbers. Find all functions
R
+
→
R
+
\mathbb{R} ^{+} \to \mathbb{R} ^{+}
R
+
→
R
+
such that
f
(
x
+
f
(
y
)
)
=
y
f
(
x
y
+
1
)
f(x+f(y)) = yf(xy+1)
f
(
x
+
f
(
y
))
=
y
f
(
x
y
+
1
)
holds for all
x
,
y
∈
R
+
x, y \in \mathbb{R} ^{+}
x
,
y
∈
R
+
.
3 variable equation
Find all triplets
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of real numbers such that
2
x
3
+
1
=
3
z
x
2x^3 + 1 = 3zx
2
x
3
+
1
=
3
z
x
2
y
3
+
1
=
3
x
y
2y^3 + 1 = 3xy
2
y
3
+
1
=
3
x
y
2
z
3
+
1
=
3
y
z
2z^3 + 1 = 3yz
2
z
3
+
1
=
3
yz