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International Contests
Czech-Polish-Slovak Match
2019 Czech-Austrian-Polish-Slovak Match
2019 Czech-Austrian-Polish-Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(6)
6
1
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Four midpoints are cyclic
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
<
A
C
AB<AC
A
B
<
A
C
and
∠
B
A
C
=
6
0
∘
\angle BAC=60^{\circ}
∠
B
A
C
=
6
0
∘
. Denote its altitudes by
A
D
,
B
E
,
C
F
AD,BE,CF
A
D
,
BE
,
CF
and its orthocenter by
H
H
H
. Let
K
,
L
,
M
K,L,M
K
,
L
,
M
be the midpoints of sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
, respectively. Prove that the midpoints of segments
A
H
,
D
K
,
E
L
,
F
M
AH, DK, EL, FM
A
H
,
DK
,
E
L
,
FM
lie on a single circle.
5
1
Hide problems
Existence of a 100 disks
Determine whether there exist
100
100
100
disks
D
2
,
D
3
,
…
,
D
101
D_2,D_3,\ldots ,D_{101}
D
2
,
D
3
,
…
,
D
101
in the plane such that the following conditions hold for all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of indices satisfying
2
≤
a
<
b
≤
101
2\le a< b\le 101
2
≤
a
<
b
≤
101
:[*] If
a
∣
b
a|b
a
∣
b
then
D
a
D_a
D
a
is contained in
D
b
D_b
D
b
. [*] If
gcd
(
a
,
b
)
=
1
\gcd (a,b)=1
g
cd
(
a
,
b
)
=
1
then
D
a
D_a
D
a
and
D
b
D_b
D
b
are disjoint.(A disk
D
(
O
,
r
)
D(O,r)
D
(
O
,
r
)
is a set of points in the plane whose distance to a given point
O
O
O
is at most a given positive real number
r
r
r
.)
3
1
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Placing arrows to form cycles
A dissection of a convex polygon into finitely many triangles by segments is called a trilateration if no three vertices of the created triangles lie on a single line (vertices of some triangles might lie inside the polygon). We say that a trilateration is good if its segments can be replaced with one-way arrows in such a way that the arrows along every triangle of the trilateration form a cycle and the arrows along the whole convex polygon also form a cycle. Find all
n
≥
3
n\ge 3
n
≥
3
such that the regular
n
n
n
-gon has a good trilateration.
2
1
Hide problems
Sum of squares of its divisors
We consider positive integers
n
n
n
having at least six positive divisors. Let the positive divisors of
n
n
n
be arranged in a sequence
(
d
i
)
1
≤
i
≤
k
(d_i)_{1\le i\le k}
(
d
i
)
1
≤
i
≤
k
with 1=d_1
n
n
n
such that
n
=
d
5
2
+
d
6
2
.
n=d_5^2+d_6^2.
n
=
d
5
2
+
d
6
2
.
1
1
Hide problems
Prove B,E,F,Y cyclic
Let
ω
\omega
ω
be a circle. Points
A
,
B
,
C
,
X
,
D
,
Y
A,B,C,X,D,Y
A
,
B
,
C
,
X
,
D
,
Y
lie on
ω
\omega
ω
in this order such that
B
D
BD
B
D
is its diameter and
D
X
=
D
Y
=
D
P
DX=DY=DP
D
X
=
D
Y
=
D
P
, where
P
P
P
is the intersection of
A
C
AC
A
C
and
B
D
BD
B
D
. Denote by
E
,
F
E,F
E
,
F
the intersections of line
X
P
XP
XP
with lines
A
B
,
B
C
AB,BC
A
B
,
BC
, respectively. Prove that points
B
,
E
,
F
,
Y
B,E,F,Y
B
,
E
,
F
,
Y
lie on a single circle.
4
1
Hide problems
all pairs of functions
Given a real number
α
\alpha
α
, find all pairs
(
f
,
g
)
(f,g)
(
f
,
g
)
of functions
f
,
g
:
R
→
R
f,g :\mathbb{R} \to \mathbb{R}
f
,
g
:
R
→
R
such that
x
f
(
x
+
y
)
+
α
⋅
y
f
(
x
−
y
)
=
g
(
x
)
+
g
(
y
)
,
∀
x
,
y
∈
R
.
xf(x+y)+\alpha \cdot yf(x-y)=g(x)+g(y) \;\;\;\;\;\;\;\;\;\;\; ,\forall x,y \in \mathbb{R}.
x
f
(
x
+
y
)
+
α
⋅
y
f
(
x
−
y
)
=
g
(
x
)
+
g
(
y
)
,
∀
x
,
y
∈
R
.