MathDB
Problems
Contests
International Contests
Middle European Mathematical Olympiad
2020 Middle European Mathematical Olympiad
2020 Middle European Mathematical Olympiad
Part of
Middle European Mathematical Olympiad
Subcontests
(4)
4#
1
Hide problems
finding n such that 1 is a sum of fractions
Find all positive integers
n
n
n
for which there exist positive integers
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \dots, x_n
x
1
,
x
2
,
…
,
x
n
such that
1
x
1
2
+
2
x
2
2
+
2
2
x
3
2
+
⋯
+
2
n
−
1
x
n
2
=
1.
\frac{1}{x_1^2}+\frac{2}{x_2^2}+\frac{2^2}{x_3^2}+\cdots +\frac{2^{n-1}}{x_n^2}=1.
x
1
2
1
+
x
2
2
2
+
x
3
2
2
2
+
⋯
+
x
n
2
2
n
−
1
=
1.
3#
1
Hide problems
exists a circle tangent to three others
Let
A
B
C
ABC
A
BC
be an acute scalene triangle with circumcircle
ω
\omega
ω
and incenter
I
I
I
. Suppose the orthocenter
H
H
H
of
B
I
C
BIC
B
I
C
lies inside
ω
\omega
ω
. Let
M
M
M
be the midpoint of the longer arc
B
C
BC
BC
of
ω
\omega
ω
. Let
N
N
N
be the midpoint of the shorter arc
A
M
AM
A
M
of
ω
\omega
ω
. Prove that there exists a circle tangent to
ω
\omega
ω
at
N
N
N
and tangent to the circumcircles of
B
H
I
BHI
B
H
I
and
C
H
I
CHI
C
H
I
.
2#
1
Hide problems
sum of digits of consecutive numbers
We call a positive integer
N
N
N
contagious if there are
1000
1000
1000
consecutive non-negative integers such that the sum of all their digits is
N
N
N
. Find all contagious positive integers.
1#
1
Hide problems
for what $k$ there are functions $f$,$g$
Let
N
\mathbb{N}
N
be the set of positive integers. Determine all positive integers
k
k
k
for which there exist functions
f
:
N
→
N
f:\mathbb{N} \to \mathbb{N}
f
:
N
→
N
and
g
:
N
→
N
g: \mathbb{N}\to \mathbb{N}
g
:
N
→
N
such that
g
g
g
assumes infinitely many values and such that
f
g
(
n
)
(
n
)
=
f
(
n
)
+
k
f^{g(n)}(n)=f(n)+k
f
g
(
n
)
(
n
)
=
f
(
n
)
+
k
holds for every positive integer
n
n
n
.(Remark. Here,
f
i
f^{i}
f
i
denotes the function
f
f
f
applied
i
i
i
times i.e
f
i
(
j
)
=
f
(
f
(
…
f
(
j
)
…
)
)
f^{i}(j)=f(f(\dots f(j)\dots ))
f
i
(
j
)
=
f
(
f
(
…
f
(
j
)
…
))
.)