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International Contests
Czech-Polish-Slovak Match
2020 Czech-Austrian-Polish-Slovak Match
2020 Czech-Austrian-Polish-Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(6)
1
1
Hide problems
collinear wanted, ABCD parallllogram tangents of circumcircle of MAD,MBC
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram whose diagonals meet at
P
P
P
. Denote by
M
M
M
the midpoint of
A
B
AB
A
B
. Let
Q
Q
Q
be a point such that
Q
A
QA
Q
A
is tangent to the circumcircle of
M
A
D
MAD
M
A
D
and
Q
B
QB
QB
is tangent to the circumcircle of
M
B
C
MBC
MBC
. Prove that points
Q
,
M
,
P
Q,M,P
Q
,
M
,
P
are collinear. (Patrik Bak, Slovakia)
6
1
Hide problems
circumcircle of XYZ passes through fixed point, <XPY = 2<BAC
Let
A
B
C
ABC
A
BC
be an acute triangle. Let
P
P
P
be a point such that
P
B
PB
PB
and
P
C
PC
PC
are tangent to circumcircle of
A
B
C
ABC
A
BC
. Let
X
X
X
and
Y
Y
Y
be variable points on
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, such that
∠
X
P
Y
=
2
∠
B
A
C
\angle XPY = 2\angle BAC
∠
XP
Y
=
2∠
B
A
C
and
P
P
P
lies in the interior of triangle
A
X
Y
AXY
A
X
Y
. Let
Z
Z
Z
be the reflection of
A
A
A
across
X
Y
XY
X
Y
. Prove that the circumcircle of
X
Y
Z
XYZ
X
Y
Z
passes through a fixed point. (Dominik Burek, Poland)
5
1
Hide problems
(x+1)^2-xy(2x-xy+2y)+(y+1)^2=n, smallest n such no of ordered pairs d(n)=61
Let
n
n
n
be a positive integer and let
d
(
n
)
d(n)
d
(
n
)
denote the number of ordered pairs of positive integers
(
x
,
y
)
(x,y)
(
x
,
y
)
such that
(
x
+
1
)
2
−
x
y
(
2
x
−
x
y
+
2
y
)
+
(
y
+
1
)
2
=
n
(x+1)^2-xy(2x-xy+2y)+(y+1)^2=n
(
x
+
1
)
2
−
x
y
(
2
x
−
x
y
+
2
y
)
+
(
y
+
1
)
2
=
n
. Find the smallest positive integer
n
n
n
satisfying
d
(
n
)
=
61
d(n) = 61
d
(
n
)
=
61
. (Patrik Bak, Slovakia)
4
1
Hide problems
(x+y)(f(x)-f(y))=a(x-y)f(x+y) where a given real
Let
a
a
a
be a given real number. Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that
(
x
+
y
)
(
f
(
x
)
−
f
(
y
)
)
=
a
(
x
−
y
)
f
(
x
+
y
)
(x+y)(f(x)-f(y))=a(x-y)f(x+y)
(
x
+
y
)
(
f
(
x
)
−
f
(
y
))
=
a
(
x
−
y
)
f
(
x
+
y
)
holds for all
x
,
y
∈
R
x,y \in R
x
,
y
∈
R
. (Walther Janous, Austria)
3
1
Hide problems
2player game connecting with line segments 1,2,...,2020 on blackboard
The numbers
1
,
2
,
.
.
.
,
2020
1, 2,..., 2020
1
,
2
,
...
,
2020
are written on the blackboard. Venus and Serena play the following game. First, Venus connects by a line segment two numbers such that one of them divides the other. Then Serena connects by a line segment two numbers which has not been connected and such that one of them divides the other. Then Venus again and they continue until there is a triangle with one vertex in
2020
2020
2020
, i.e.
2020
2020
2020
is connected to two numbers that are connected with each other. The girl that has drawn the last line segment (completed the triangle) is the winner. Which of the girls has a winning strategy? (Tomáš Bárta, Czech Republic)
2
1
Hide problems
[a,b] contains infinitely many 2020-good numbers, real x= sum 1/a_i, a_i in N
Given a positive integer
n
n
n
, we say that a real number
x
x
x
is
n
n
n
-good if there exist
n
n
n
positive integers
a
1
,
.
.
.
,
a
n
a_1,...,a_n
a
1
,
...
,
a
n
such that
x
=
1
a
1
+
.
.
.
+
1
a
n
.
x=\frac{1}{a_1}+...+\frac{1}{a_n}.
x
=
a
1
1
+
...
+
a
n
1
.
Find all positive integers
k
k
k
for which the following assertion is true: if
a
,
b
a,b
a
,
b
are real numbers such that the closed interval
[
a
,
b
]
[a,b]
[
a
,
b
]
contains infinitely many
2020
2020
2020
-good numbers, then the interval
[
a
,
b
]
[a,b]
[
a
,
b
]
contains at least one
k
k
k
-good number.(Josef Tkadlec, Czech Republic)