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Problems
Contests
International Contests
Middle European Mathematical Olympiad
2017 Middle European Mathematical Olympiad
2017 Middle European Mathematical Olympiad
Part of
Middle European Mathematical Olympiad
Subcontests
(8)
8
1
Hide problems
Prime factorization exponents form a geometric progression
For an integer
n
≥
3
n \geq 3
n
≥
3
we define the sequence
α
1
,
α
2
,
…
,
α
k
\alpha_1, \alpha_2, \ldots, \alpha_k
α
1
,
α
2
,
…
,
α
k
as the sequence of exponents in the prime factorization of
n
!
=
p
1
α
1
p
2
α
2
…
p
k
α
k
n! = p_1^{\alpha_1}p_2^{\alpha_2} \ldots p_k^{\alpha_k}
n
!
=
p
1
α
1
p
2
α
2
…
p
k
α
k
, where
p
1
<
p
2
<
…
<
p
k
p_1 < p_2 < \ldots < p_k
p
1
<
p
2
<
…
<
p
k
are primes. Determine all integers
n
≥
3
n \geq 3
n
≥
3
for which
α
1
,
α
2
,
…
,
α
k
\alpha_1, \alpha_2, \ldots, \alpha_k
α
1
,
α
2
,
…
,
α
k
is a geometric progression.
7
1
Hide problems
Partial sums are distinct modulo n
Determine all integers
n
≥
2
n \geq 2
n
≥
2
such that there exists a permutation
x
0
,
x
1
,
…
,
x
n
−
1
x_0, x_1, \ldots, x_{n - 1}
x
0
,
x
1
,
…
,
x
n
−
1
of the numbers
0
,
1
,
…
,
n
−
1
0, 1, \ldots, n - 1
0
,
1
,
…
,
n
−
1
with the property that the
n
n
n
numbers
x
0
,
x
0
+
x
1
,
…
,
x
0
+
x
1
+
…
+
x
n
−
1
x_0, \hspace{0.3cm} x_0 + x_1, \hspace{0.3cm} \ldots, \hspace{0.3cm} x_0 + x_1 + \ldots + x_{n - 1}
x
0
,
x
0
+
x
1
,
…
,
x
0
+
x
1
+
…
+
x
n
−
1
are pairwise distinct modulo
n
n
n
.
6
1
Hide problems
Fairly standard configuration involving tangents
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
B
≠
A
C
AB \neq AC
A
B
=
A
C
, circumcentre
O
O
O
and circumcircle
Γ
\Gamma
Γ
. Let the tangents to
Γ
\Gamma
Γ
at
B
B
B
and
C
C
C
meet each other at
D
D
D
, and let the line
A
O
AO
A
O
intersect
B
C
BC
BC
at
E
E
E
. Denote the midpoint of
B
C
BC
BC
by
M
M
M
and let
A
M
AM
A
M
meet
Γ
\Gamma
Γ
again at
N
≠
A
N \neq A
N
=
A
. Finally, let
F
≠
A
F \neq A
F
=
A
be a point on
Γ
\Gamma
Γ
such that
A
,
M
,
E
A, M, E
A
,
M
,
E
and
F
F
F
are concyclic. Prove that
F
N
FN
FN
bisects the segment
M
D
MD
M
D
.
5
1
Hide problems
Midpoint of the segment joining two excentres of a triangle
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
B
>
A
C
AB > AC
A
B
>
A
C
and circumcircle
Γ
\Gamma
Γ
. Let
M
M
M
be the midpoint of the shorter arc
B
C
BC
BC
of
Γ
\Gamma
Γ
, and let
D
D
D
be the intersection of the rays
A
C
AC
A
C
and
B
M
BM
BM
. Let
E
≠
C
E \neq C
E
=
C
be the intersection of the internal bisector of the angle
A
C
B
ACB
A
CB
and the circumcircle of the triangle
B
D
C
BDC
B
D
C
. Let us assume that
E
E
E
is inside the triangle
A
B
C
ABC
A
BC
and there is an intersection
N
N
N
of the line
D
E
DE
D
E
and the circle
Γ
\Gamma
Γ
such that
E
E
E
is the midpoint of the segment
D
N
DN
D
N
. Show that
N
N
N
is the midpoint of the segment
I
B
I
C
I_B I_C
I
B
I
C
, where
I
B
I_B
I
B
and
I
C
I_C
I
C
are the excentres of
A
B
C
ABC
A
BC
opposite to
B
B
B
and
C
C
C
, respectively.
4
2
Hide problems
Minimum absolute difference of powers of 2 and 181
Determine the smallest possible value of
∣
2
m
−
18
1
n
∣
,
|2^m - 181^n|,
∣
2
m
−
18
1
n
∣
,
where
m
m
m
and
n
n
n
are positive integers.
Maximum number of ccw oriented polylines
Let
n
≥
3
n \geq 3
n
≥
3
be an integer. A sequence
P
1
,
P
2
,
…
,
P
n
P_1, P_2, \ldots, P_n
P
1
,
P
2
,
…
,
P
n
of distinct points in the plane is called good if no three of them are collinear, the polyline
P
1
P
2
…
P
n
P_1P_2 \ldots P_n
P
1
P
2
…
P
n
is non-self-intersecting and the triangle
P
i
P
i
+
1
P
i
+
2
P_iP_{i + 1}P_{i + 2}
P
i
P
i
+
1
P
i
+
2
is oriented counterclockwise for every
i
=
1
,
2
,
…
,
n
−
2
i = 1, 2, \ldots, n - 2
i
=
1
,
2
,
…
,
n
−
2
. For every integer
n
≥
3
n \geq 3
n
≥
3
determine the greatest possible integer
k
k
k
with the following property: there exist
n
n
n
distinct points
A
1
,
A
2
,
…
,
A
n
A_1, A_2, \ldots, A_n
A
1
,
A
2
,
…
,
A
n
in the plane for which there are
k
k
k
distinct permutations
σ
:
{
1
,
2
,
…
,
n
}
→
{
1
,
2
,
…
,
n
}
\sigma : \{1, 2, \ldots, n\} \to \{1, 2, \ldots, n\}
σ
:
{
1
,
2
,
…
,
n
}
→
{
1
,
2
,
…
,
n
}
such that
A
σ
(
1
)
,
A
σ
(
2
)
,
…
,
A
σ
(
n
)
A_{\sigma(1)}, A_{\sigma(2)}, \ldots, A_{\sigma(n)}
A
σ
(
1
)
,
A
σ
(
2
)
,
…
,
A
σ
(
n
)
is good. (A polyline
P
1
P
2
…
P
n
P_1P_2 \ldots P_n
P
1
P
2
…
P
n
consists of the segments
P
1
P
2
,
P
2
P
3
,
…
,
P
n
−
1
P
n
P_1P_2, P_2P_3, \ldots, P_{n - 1}P_n
P
1
P
2
,
P
2
P
3
,
…
,
P
n
−
1
P
n
.)
3
2
Hide problems
Minimum number of bad lamps on a board
There is a lamp on each cell of a
2017
×
2017
2017 \times 2017
2017
×
2017
board. Each lamp is either on or off. A lamp is called bad if it has an even number of neighbours that are on. What is the smallest possible number of bad lamps on such a board? (Two lamps are neighbours if their respective cells share a side.)
Convex pentagon
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a convex pentagon. Let
P
P
P
be the intersection of the lines
C
E
CE
CE
and
B
D
BD
B
D
. Assume that
∠
P
A
D
=
∠
A
C
B
\angle PAD = \angle ACB
∠
P
A
D
=
∠
A
CB
and
∠
C
A
P
=
∠
E
D
A
\angle CAP = \angle EDA
∠
C
A
P
=
∠
E
D
A
. Prove that the circumcentres of the triangles
A
B
C
ABC
A
BC
and
A
D
E
ADE
A
D
E
are collinear with
P
P
P
.
2
2
Hide problems
Inequality in three variables with sum -1
Determine the smallest possible real constant
C
C
C
such that the inequality
∣
x
3
+
y
3
+
z
3
+
1
∣
≤
C
∣
x
5
+
y
5
+
z
5
+
1
∣
|x^3 + y^3 + z^3 + 1| \leq C|x^5 + y^5 + z^5 + 1|
∣
x
3
+
y
3
+
z
3
+
1∣
≤
C
∣
x
5
+
y
5
+
z
5
+
1∣
holds for all real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
satisfying
x
+
y
+
z
=
−
1
x + y + z = -1
x
+
y
+
z
=
−
1
.
Memorable labellings of a regular n-gon
Let
n
≥
3
n \geq 3
n
≥
3
be an integer. A labelling of the
n
n
n
vertices, the
n
n
n
sides and the interior of a regular
n
n
n
-gon by
2
n
+
1
2n + 1
2
n
+
1
distinct integers is called memorable if the following conditions hold: (a) Each side has a label that is the arithmetic mean of the labels of its endpoints. (b) The interior of the
n
n
n
-gon has a label that is the arithmetic mean of the labels of all the vertices. Determine all integers
n
≥
3
n \geq 3
n
≥
3
for which there exists a memorable labelling of a regular
n
n
n
-gon consisting of
2
n
+
1
2n + 1
2
n
+
1
consecutive integers.
1
2
Hide problems
Polynomial FE
Determine all pairs of polynomials
(
P
,
Q
)
(P, Q)
(
P
,
Q
)
with real coefficients satisfying
P
(
x
+
Q
(
y
)
)
=
Q
(
x
+
P
(
y
)
)
P(x + Q(y)) = Q(x + P(y))
P
(
x
+
Q
(
y
))
=
Q
(
x
+
P
(
y
))
for all real numbers
x
x
x
and
y
y
y
.
Functional equation on the set of reals
Determine all functions
f
:
R
→
R
f : \mathbb{R} \to \mathbb{R}
f
:
R
→
R
satisfying
f
(
x
2
+
f
(
x
)
f
(
y
)
)
=
x
f
(
x
+
y
)
f(x^2 + f(x)f(y)) = xf(x + y)
f
(
x
2
+
f
(
x
)
f
(
y
))
=
x
f
(
x
+
y
)
for all real numbers
x
x
x
and
y
y
y
.