MathDB
Problems
Contests
International Contests
Middle European Mathematical Olympiad
2023 Middle European Mathematical Olympiad
2023 Middle European Mathematical Olympiad
Part of
Middle European Mathematical Olympiad
Subcontests
(8)
8
1
Hide problems
Gcd sequence
Let
A
,
B
∈
N
A, B \in \mathbb{N}
A
,
B
∈
N
. Consider a sequence
x
1
,
x
2
,
…
x_1, x_2, \ldots
x
1
,
x
2
,
…
such that for all
n
≥
2
n\geq 2
n
≥
2
,
x
n
+
1
=
A
⋅
gcd
(
x
n
,
x
n
−
1
)
+
B
.
x_{n+1}=A \cdot \gcd(x_n, x_{n-1})+B.
x
n
+
1
=
A
⋅
g
cd
(
x
n
,
x
n
−
1
)
+
B
.
Show that the sequence attains only finitely many distinct values.
7
1
Hide problems
Representable integers
Find all positive integers
n
n
n
, for which there exist positive integers
a
>
b
a>b
a
>
b
, satisfying
n
=
4
a
b
a
−
b
n=\frac{4ab}{a-b}
n
=
a
−
b
4
ab
.
6
1
Hide problems
Hard excircle geo
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
<
A
C
AB < AC
A
B
<
A
C
. Let
J
J
J
be the center of the
A
A
A
-excircle of
A
B
C
ABC
A
BC
. Let
D
D
D
be the projection of
J
J
J
on line
B
C
BC
BC
. The internal bisectors of angles
B
D
J
BDJ
B
D
J
and
J
D
C
JDC
J
D
C
intersectlines
B
J
BJ
B
J
and
J
C
JC
J
C
at
X
X
X
and
Y
Y
Y
, respectively. Segments
X
Y
XY
X
Y
and
J
D
JD
J
D
intersect at
P
P
P
. Let
Q
Q
Q
be the projection of
A
A
A
on line
B
C
BC
BC
. Prove that the internal angle bisector of
Q
A
P
QAP
Q
A
P
is perpendicular to line
X
Y
XY
X
Y
.Proposed by Dominik Burek, Poland
5
1
Hide problems
Quadrilateral geo with a condition
We are given a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
whose angles are not right. Assume there are points
P
,
Q
,
R
,
S
P, Q, R, S
P
,
Q
,
R
,
S
on its sides
A
B
,
B
C
,
C
D
,
D
A
AB, BC, CD, DA
A
B
,
BC
,
C
D
,
D
A
, respectively, such that
P
S
∥
B
D
PS \parallel BD
PS
∥
B
D
,
S
Q
⊥
B
C
SQ \perp BC
SQ
⊥
BC
,
P
R
⊥
C
D
PR \perp CD
PR
⊥
C
D
. Furthermore, assume that the lines
P
R
,
S
Q
PR, SQ
PR
,
SQ
, and
A
C
AC
A
C
are concurrent. Prove thatthe points
P
,
Q
,
R
,
S
P, Q, R, S
P
,
Q
,
R
,
S
are concyclic.
4
2
Hide problems
(n, m)-good sets
Let
n
,
m
n, m
n
,
m
be positive integers. A set
S
S
S
of positive integers is called
(
n
,
m
)
(n, m)
(
n
,
m
)
-good, if:(1)
m
∈
S
m \in S
m
∈
S
; (2) for all
a
∈
S
a\in S
a
∈
S
, all divisors of
a
a
a
are also in
S
S
S
; (3) for all distinct
a
,
b
∈
S
a, b \in S
a
,
b
∈
S
,
a
n
+
b
n
∈
S
a^n+b^n \in S
a
n
+
b
n
∈
S
.For which
(
n
,
m
)
(n, m)
(
n
,
m
)
, the only
(
n
,
m
)
(n, m)
(
n
,
m
)
-good set is
N
\mathbb{N}
N
?
Teams with non-clashing uniforms
Let
c
≥
4
c \geq 4
c
≥
4
be an even integer. In some football league, each team has a home uniform and anaway uniform. Every home uniform is coloured in two different colours, and every away uniformis coloured in one colour. A team’s away uniform cannot be coloured in one of the colours fromthe home uniform. There are at most
c
c
c
distinct colours on all of the uniforms. If two teams havethe same two colours on their home uniforms, then they have different colours on their away uniforms. We say a pair of uniforms is clashing if some colour appears on both of them. Suppose that for every team
X
X
X
in the league, there is no team
Y
Y
Y
in the league such that the home uniform of
X
X
X
is clashing with both uniforms of
Y
Y
Y
. Determine the maximum possible number of teams in the league.
3
2
Hide problems
Easy incircle geo
Let
A
B
C
ABC
A
BC
be a triangle with incenter
I
I
I
, and the incircle touches
B
C
BC
BC
at
D
D
D
. The points
E
,
F
E, F
E
,
F
are such that
B
E
∥
A
I
∥
C
F
BE \parallel AI \parallel CF
BE
∥
A
I
∥
CF
and
∠
B
E
I
=
∠
C
F
I
=
9
0
∘
\angle BEI=\angle CFI=90^{\circ}
∠
BE
I
=
∠
CF
I
=
9
0
∘
. If
D
E
,
D
F
DE, DF
D
E
,
D
F
meet the incircle at
E
′
,
F
′
E', F'
E
′
,
F
′
, show that
E
′
F
′
⊥
A
I
E'F' \perp AI
E
′
F
′
⊥
A
I
.
Bishops on a chessboard
Find the smallest integer
b
b
b
with the following property: For each way of colouring exactly
b
b
b
squares of an
8
×
8
8 \times 8
8
×
8
chessboard green, one can place
7
7
7
bishops on
7
7
7
green squares so that no two bishops attack each other.
2
2
Hide problems
Chords in a circle
Find all positive integers
n
≥
3
n \geq 3
n
≥
3
, for which it is possible to draw
n
n
n
chords on a circle, with their
2
n
2n
2
n
endpoints being pairwise distinct, such that each chords intersects exactly
k
k
k
others for:(a)
k
=
n
−
2
k=n-2
k
=
n
−
2
,(b)
k
=
n
−
3
k=n-3
k
=
n
−
3
.
4-variable inequality
If
a
,
b
,
c
,
d
>
0
a, b, c, d>0
a
,
b
,
c
,
d
>
0
and
a
b
c
d
=
1
abcd=1
ab
c
d
=
1
, show that
a
b
+
1
a
+
1
+
b
c
+
1
b
+
1
+
c
d
+
1
c
+
1
+
d
a
+
1
d
+
1
≥
4.
\frac{ab+1}{a+1}+\frac{bc+1}{b+1}+\frac{cd+1}{c+1}+\frac{da+1}{d+1} \geq 4.
a
+
1
ab
+
1
+
b
+
1
b
c
+
1
+
c
+
1
c
d
+
1
+
d
+
1
d
a
+
1
≥
4.
When does equality hold?
1
2
Hide problems
Functional inequality
For each pair
(
α
,
β
)
(\alpha, \beta)
(
α
,
β
)
of non-negative reals with
α
+
β
≥
2
\alpha+\beta \geq 2
α
+
β
≥
2
, determine all functions
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
, such that
f
(
x
)
f
(
y
)
≤
f
(
x
y
)
+
α
x
+
β
y
f(x)f(y) \leq f(xy)+\alpha x+\beta y
f
(
x
)
f
(
y
)
≤
f
(
x
y
)
+
αx
+
β
y
for all reals
x
,
y
x, y
x
,
y
.
Maximal number of distinct values
(a) A function
f
:
Z
→
Z
f:\mathbb{Z} \rightarrow \mathbb{Z}
f
:
Z
→
Z
is called
Z
\mathbb{Z}
Z
-good if
f
(
a
2
+
b
)
=
f
(
b
2
+
a
)
f(a^2+b)=f(b^2+a)
f
(
a
2
+
b
)
=
f
(
b
2
+
a
)
for all
a
,
b
∈
Z
a, b \in \mathbb{Z}
a
,
b
∈
Z
. What is the largest possible number of distinct values that can occur among
f
(
1
)
,
…
,
f
(
2023
)
f(1), \ldots, f(2023)
f
(
1
)
,
…
,
f
(
2023
)
, where
f
f
f
is a
Z
\mathbb{Z}
Z
-good function?(b) A function
f
:
N
→
N
f:\mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
is called
N
\mathbb{N}
N
-good if
f
(
a
2
+
b
)
=
f
(
b
2
+
a
)
f(a^2+b)=f(b^2+a)
f
(
a
2
+
b
)
=
f
(
b
2
+
a
)
for all
a
,
b
∈
N
a, b \in \mathbb{N}
a
,
b
∈
N
. What is the largest possible number of distinct values that can occur among
f
(
1
)
,
…
,
f
(
2023
)
f(1), \ldots, f(2023)
f
(
1
)
,
…
,
f
(
2023
)
, where
f
f
f
is a
N
\mathbb{N}
N
-good function?