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Problems
Contests
International Contests
Middle European Mathematical Olympiad
2021 Middle European Mathematical Olympiad
2021 Middle European Mathematical Olympiad
Part of
Middle European Mathematical Olympiad
Subcontests
(8)
8
1
Hide problems
Perfect squares in base 4 with only digits 1 and 2
Prove that there are infinitely many positive integers
n
n
n
such that
n
2
n^2
n
2
written in base
4
4
4
contains only digits
1
1
1
and
2
2
2
.
7
1
Hide problems
Diophantine equation with primes
Find all pairs
(
n
,
p
)
(n, p)
(
n
,
p
)
of positive integers such that
p
p
p
is prime and
1
+
2
+
⋯
+
n
=
3
⋅
(
1
2
+
2
2
+
⋅
+
p
2
)
.
1 + 2 + \cdots + n = 3 \cdot (1^2 + 2^2 + \cdot + p^2).
1
+
2
+
⋯
+
n
=
3
⋅
(
1
2
+
2
2
+
⋅
+
p
2
)
.
6
1
Hide problems
Length bash for the win
Let
A
B
C
ABC
A
BC
be a triangle and let
M
M
M
be the midpoint of the segment
B
C
BC
BC
. Let
X
X
X
be a point on the ray
A
B
AB
A
B
such that
2
∠
C
X
A
=
∠
C
M
A
2 \angle CXA=\angle CMA
2∠
CX
A
=
∠
CM
A
. Let
Y
Y
Y
be a point on the ray
A
C
AC
A
C
such that
2
∠
A
Y
B
=
∠
A
M
B
2 \angle AYB=\angle AMB
2∠
A
Y
B
=
∠
A
MB
. The line
B
C
BC
BC
intersects the circumcircle of the triangle
A
X
Y
AXY
A
X
Y
at
P
P
P
and
Q
Q
Q
, such that the points
P
,
B
,
C
P, B, C
P
,
B
,
C
, and
Q
Q
Q
lie in this order on the line
B
C
BC
BC
. Prove that
P
B
=
Q
C
PB=QC
PB
=
QC
.Proposed by Dominik Burek, Poland
5
1
Hide problems
Simple geometry
Let
A
D
AD
A
D
be the diameter of the circumcircle of an acute triangle
A
B
C
ABC
A
BC
. The lines through
D
D
D
parallel to
A
B
AB
A
B
and
A
C
AC
A
C
meet lines
A
C
AC
A
C
and
A
B
AB
A
B
in points
E
E
E
and
F
F
F
, respectively. Lines
E
F
EF
EF
and
B
C
BC
BC
meet at
G
G
G
. Prove that
A
D
AD
A
D
and
D
G
DG
D
G
are perpendicular.
4
2
Hide problems
Zagi jumping on a polygon
Let
n
≥
3
n \ge 3
n
≥
3
be an integer. Zagi the squirrel sits at a vertex of a regular
n
n
n
-gon. Zagi plans to make a journey of
n
−
1
n-1
n
−
1
jumps such that in the
i
i
i
-th jump, it jumps by
i
i
i
edges clockwise, for
i
∈
{
1
,
…
,
n
−
1
}
i \in \{1, \ldots,n-1 \}
i
∈
{
1
,
…
,
n
−
1
}
. Prove that if after
⌈
n
2
⌉
\lceil \tfrac{n}{2} \rceil
⌈
2
n
⌉
jumps Zagi has visited
⌈
n
2
⌉
+
1
\lceil \tfrac{n}{2} \rceil+1
⌈
2
n
⌉
+
1
distinct vertices, then after
n
−
1
n-1
n
−
1
jumps Zagi will have visited all of the vertices. (Remark. For a real number
x
x
x
, we denote by
⌈
x
⌉
\lceil x \rceil
⌈
x
⌉
the smallest integer larger or equal to
x
x
x
.)
Diagonals are concurrent triplewise
Let
n
n
n
be a positive integer. Prove that in a regular
6
n
6n
6
n
-gon, we can draw
3
n
3n
3
n
diagonals with pairwise distinct ends and partition the drawn diagonals into
n
n
n
triplets so that:[*] the diagonals in each triplet intersect in one interior point of the polygon and [*] all these
n
n
n
intersection points are distinct.
3
2
Hide problems
Easy Geometry
Let
A
B
C
ABC
A
BC
be an acute triangle and
D
D
D
an interior point of segment
B
C
BC
BC
. Points
E
E
E
and
F
F
F
lie in the half-plane determined by the line
B
C
BC
BC
containing
A
A
A
such that
D
E
DE
D
E
is perpendicular to
B
E
BE
BE
and
D
E
DE
D
E
is tangent to the circumcircle of
A
C
D
ACD
A
C
D
, while
D
F
DF
D
F
is perpendicular to
C
F
CF
CF
and
D
F
DF
D
F
is tangent to the circumcircle of
A
B
D
ABD
A
B
D
. Prove that the points
A
,
D
,
E
A, D, E
A
,
D
,
E
and
F
F
F
are concyclic.
Captain jack distribute coins equally
Let
n
,
b
n, b
n
,
b
and
c
c
c
be positive integers. A group of
n
n
n
pirates wants to fairly split their treasure. The treasure consists of
c
⋅
n
c \cdot n
c
⋅
n
identical coins distributed over
b
⋅
n
b \cdot n
b
⋅
n
bags, of which at least
n
−
1
n-1
n
−
1
bags are initially empty. Captain Jack inspects the contents of each bag and then performs a sequence of moves. In one move, he can take any number of coins from a single bag and put them into one empty bag. Prove that no matter how the coins are initially distributed, Jack can perform at most
n
−
1
n-1
n
−
1
moves and then split the bags among the pirates such that each pirate gets
b
b
b
bags and
c
c
c
coins.
2
2
Hide problems
Bishop circuit
Let
m
m
m
and
n
n
n
be positive integers. Some squares of an
m
×
n
m \times n
m
×
n
board are coloured red. A sequence
a
1
,
a
2
,
…
,
a
2
r
a_1, a_2, \ldots , a_{2r}
a
1
,
a
2
,
…
,
a
2
r
of
2
r
≥
4
2r \ge 4
2
r
≥
4
pairwise distinct red squares is called a bishop circuit if for every
k
∈
{
1
,
…
,
2
r
}
k \in \{1, \ldots , 2r \}
k
∈
{
1
,
…
,
2
r
}
, the squares
a
k
a_k
a
k
and
a
k
+
1
a_{k+1}
a
k
+
1
lie on a diagonal, but the squares
a
k
a_k
a
k
and
a
k
+
2
a_{k+2}
a
k
+
2
do not lie on a diagonal (here
a
2
r
+
1
=
a
1
a_{2r+1}=a_1
a
2
r
+
1
=
a
1
and
a
2
r
+
2
=
a
2
a_{2r+2}=a_2
a
2
r
+
2
=
a
2
). In terms of
m
m
m
and
n
n
n
, determine the maximum possible number of red squares on an
m
×
n
m \times n
m
×
n
board without a bishop circuit. (Remark. Two squares lie on a diagonal if the line passing through their centres intersects the sides of the board at an angle of
4
5
∘
45^\circ
4
5
∘
.)
Pretty Polynomials
Given a positive integer
n
n
n
, we say that a polynomial
P
P
P
with real coefficients is
n
n
n
-pretty if the equation
P
(
⌊
x
⌋
)
=
⌊
P
(
x
)
⌋
P(\lfloor x \rfloor)=\lfloor P(x) \rfloor
P
(⌊
x
⌋)
=
⌊
P
(
x
)⌋
has exactly
n
n
n
real solutions. Show that for each positive integer
n
n
n
[*] there exists an n-pretty polynomial; [*] any
n
n
n
-pretty polynomial has a degree of at least
2
n
+
1
3
\tfrac{2n+1}{3}
3
2
n
+
1
.(Remark. For a real number
x
x
x
, we denote by
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
the largest integer smaller than or equal to
x
x
x
.)
1
2
Hide problems
Simple sequences
Determine all real numbers A such that every sequence of non-zero real numbers
x
1
,
x
2
,
…
x_1, x_2, \ldots
x
1
,
x
2
,
…
satisfying
x
n
+
1
=
A
−
1
x
n
x_{n+1}=A-\frac{1}{x_n}
x
n
+
1
=
A
−
x
n
1
for every integer
n
≥
1
n \ge 1
n
≥
1
, has only finitely many negative terms.
Functional Inequality (is back)
Determine all functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that the inequality
f
(
x
2
)
−
f
(
y
2
)
≤
(
f
(
x
)
+
y
)
(
x
−
f
(
y
)
)
f(x^2)-f(y^2) \le (f(x)+y)(x-f(y))
f
(
x
2
)
−
f
(
y
2
)
≤
(
f
(
x
)
+
y
)
(
x
−
f
(
y
))
holds for all real numbers
x
x
x
and
y
y
y
.