MathDB
Problems
Contests
International Contests
Czech-Polish-Slovak Match
2016 Czech-Polish-Slovak Match
2016 Czech-Polish-Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(3)
1
2
Hide problems
Czech-Polish-Slovak Match 2016
Let
P
P
P
be a non-degenerate polygon with
n
n
n
sides, where
n
>
4
n > 4
n
>
4
. Prove that there exist three distinct vertices
A
,
B
,
C
A, B, C
A
,
B
,
C
of
P
P
P
with the following property:If
ℓ
1
,
ℓ
2
,
ℓ
3
\ell_1,\ell_2,\ell_3
ℓ
1
,
ℓ
2
,
ℓ
3
are the lengths of the three polygonal chains into which
A
,
B
,
C
A, B, C
A
,
B
,
C
break the perimeter of
P
P
P
, then there is a triangle with side lengths
ℓ
1
,
ℓ
2
\ell_1,\ell_2
ℓ
1
,
ℓ
2
and
ℓ
3
\ell_3
ℓ
3
. Remark: By a non-degenerate polygon we mean a polygon in which every two sides are disjoint, apart from consecutive ones, which share only the common endpoint.(Poland)
Czech-Polish-Slovak Match 2016
Find all quadruplets
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
of real numbers satisfying the system
(
a
+
b
)
(
a
2
+
b
2
)
=
(
c
+
d
)
(
c
2
+
d
2
)
(a + b)(a^2 + b^2) = (c + d)(c^2 + d^2)
(
a
+
b
)
(
a
2
+
b
2
)
=
(
c
+
d
)
(
c
2
+
d
2
)
(
a
+
c
)
(
a
2
+
c
2
)
=
(
b
+
d
)
(
b
2
+
d
2
)
(a + c)(a^2 + c^2) = (b + d)(b^2 + d^2)
(
a
+
c
)
(
a
2
+
c
2
)
=
(
b
+
d
)
(
b
2
+
d
2
)
(
a
+
d
)
(
a
2
+
d
2
)
=
(
b
+
c
)
(
b
2
+
c
2
)
(a + d)(a^2 + d^2) = (b + c)(b^2 + c^2)
(
a
+
d
)
(
a
2
+
d
2
)
=
(
b
+
c
)
(
b
2
+
c
2
)
(Slovakia)
2
2
Hide problems
Czech-Polish-Slovak Match 2016
Let
m
,
n
>
2
m,n > 2
m
,
n
>
2
be even integers. Consider a board of size m \times n whose every cell is colored either black or white. The Guesser does not see the coloring of the board but may ask the Oracle some questions about it. In particular, she may inquire about two adjacent cells (sharing an edge) and the Oracle discloses whether the two adjacent cells have the same color or not. The Guesser eventually wants to fi nd the parity of the number of adjacent cell-pairs whose colors are diff erent. What is the minimum number of inquiries the Guesser needs to make so that she is guaranteed to find her answer?(Czech Republic)
Czech-Polish-Slovak Match 2016
Prove that for every non-negative integer
n
n
n
there exist integers
x
,
y
,
z
x, y, z
x
,
y
,
z
with
g
c
d
(
x
,
y
,
z
)
=
1
gcd(x, y, z) = 1
g
c
d
(
x
,
y
,
z
)
=
1
, such that
x
2
+
y
2
+
z
2
=
3
2
n
x^2 + y^2 + z^2 = 3^{2^n}
x
2
+
y
2
+
z
2
=
3
2
n
.(Poland)
3
2
Hide problems
Czech-Polish-Slovak Match 2016
Let
n
n
n
be a positive integer. For a fi nite set
M
M
M
of positive integers and each
i
∈
{
0
,
1
,
.
.
.
,
n
−
1
}
i \in \{0,1,..., n-1\}
i
∈
{
0
,
1
,
...
,
n
−
1
}
, we denote
s
i
s_i
s
i
the number of non-empty subsets of
M
M
M
whose sum of elements gives remainder
i
i
i
after division by
n
n
n
. We say that
M
M
M
is "
n
n
n
-balanced" if
s
0
=
s
1
=
.
.
.
.
=
s
n
−
1
s_0 = s_1 =....= s_{n-1}
s
0
=
s
1
=
....
=
s
n
−
1
. Prove that for every odd number
n
n
n
there exists a non-empty
n
n
n
-balanced subset of
{
0
,
1
,
.
.
.
,
n
}
\{0,1,..., n\}
{
0
,
1
,
...
,
n
}
. For example if
n
=
5
n = 5
n
=
5
and
M
=
{
1
,
3
,
4
}
M = \{1,3,4\}
M
=
{
1
,
3
,
4
}
, we have
s
0
=
s
1
=
s
2
=
1
,
s
3
=
s
4
=
2
s_0 = s_1 = s_2 = 1, s_3 = s_4 = 2
s
0
=
s
1
=
s
2
=
1
,
s
3
=
s
4
=
2
so
M
M
M
is not
5
5
5
-balanced.(Czech Republic)
Czech-Polish-Slovak Match 2016
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
B
<
A
C
AB < AC
A
B
<
A
C
. Tangent to its circumcircle
Ω
\Omega
Ω
at
A
A
A
intersects the line
B
C
BC
BC
at
D
D
D
. Let
G
G
G
be the centroid of
△
A
B
C
\triangle ABC
△
A
BC
and let
A
G
AG
A
G
meet
Ω
\Omega
Ω
again at
H
≠
A
H \neq A
H
=
A
. Suppose the line
D
G
DG
D
G
intersects the lines
A
B
AB
A
B
and
A
C
AC
A
C
at
E
E
E
and
F
F
F
, respectively. Prove that
∠
E
H
G
=
∠
G
H
F
\angle EHG = \angle GHF
∠
E
H
G
=
∠
G
H
F
.(Slovakia)