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Contests
International Contests
Czech-Polish-Slovak Junior Match
2019 Czech-Polish-Slovak Junior Match
2019 Czech-Polish-Slovak Junior Match
Part of
Czech-Polish-Slovak Junior Match
Subcontests
(6)
6
1
Hide problems
cyclic ABCD and equal angles given, perpendicular wanted
Given is a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
. Points
K
,
L
,
M
,
N
K, L, M, N
K
,
L
,
M
,
N
lying on sides
A
B
,
B
C
,
C
D
,
D
A
AB, BC, CD, DA
A
B
,
BC
,
C
D
,
D
A
, respectively, satisfy
∠
A
D
K
=
∠
B
C
K
\angle ADK=\angle BCK
∠
A
DK
=
∠
BC
K
,
∠
B
A
L
=
∠
C
D
L
\angle BAL=\angle CDL
∠
B
A
L
=
∠
C
D
L
,
∠
C
B
M
=
∠
D
A
M
\angle CBM =\angle DAM
∠
CBM
=
∠
D
A
M
,
∠
D
C
N
=
∠
A
B
N
\angle DCN =\angle ABN
∠
D
CN
=
∠
A
BN
. Prove that lines
K
M
KM
K
M
and
L
N
LN
L
N
are perpendicular.
5
2
Hide problems
everyone has d friends, every 2 strangers have exactly one common friend
Given is a group in which everyone has exactly
d
d
d
friends and every two strangers have exactly one common friend. Prove that there are at most
d
2
+
1
d^2 + 1
d
2
+
1
people in this group.
120 rotations around it's center of a regular 360gon
Let
A
1
A
2
.
.
.
A
360
A_1A_2 ...A_{360}
A
1
A
2
...
A
360
be a regular
360
360
360
-gon with centre
S
S
S
. For each of the triangles
A
1
A
50
A
68
A_1A_{50}A_{68}
A
1
A
50
A
68
and
A
1
A
50
A
69
A_1A_{50}A_{69}
A
1
A
50
A
69
determine, whether its images under some
120
120
120
rotations with centre
S
S
S
can have (as triangles) all the
360
360
360
points
A
1
,
A
2
,
.
.
.
,
A
360
A_1, A_2, ..., A_{360}
A
1
,
A
2
,
...
,
A
360
as vertices.
4
2
Hide problems
similar triangles for juniors
Let
k
k
k
be a circle with diameter
A
B
AB
A
B
. A point
C
C
C
is chosen inside the segment
A
B
AB
A
B
and a point
D
D
D
is chosen on
k
k
k
such that
B
C
D
BCD
BC
D
is an acute-angled triangle, with circumcentre denoted by
O
O
O
. Let
E
E
E
be the intersection of the circle
k
k
k
and the line
B
O
BO
BO
(different from
B
B
B
). Show that the triangles
B
C
D
BCD
BC
D
and
E
C
A
ECA
EC
A
are similar.
possible values of xy+yz+zx when x^2-yz = y^2-zx = z^2-xy = 2
Determine all possible values of the expression
x
y
+
y
z
+
z
x
xy+yz+zx
x
y
+
yz
+
z
x
with real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
satisfying the conditions
x
2
−
y
z
=
y
2
−
z
x
=
z
2
−
x
y
=
2
x^2-yz = y^2-zx = z^2-xy = 2
x
2
−
yz
=
y
2
−
z
x
=
z
2
−
x
y
=
2
.
3
2
Hide problems
fill the n x n table with numbers 1, 2 and -3 so that each sum is 0
Determine all positive integers
n
n
n
such that it is possible to fill the
n
×
n
n \times n
n
×
n
table with numbers
1
,
2
1, 2
1
,
2
and
−
3
-3
−
3
so that the sum of the numbers in each row and each column is
0
0
0
.
cyclic ABCD wanted, perpendicular diagonals given
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with perpendicular diagonals, such that
∠
B
A
C
=
∠
A
D
B
\angle BAC = \angle ADB
∠
B
A
C
=
∠
A
D
B
,
∠
C
B
D
=
∠
D
C
A
\angle CBD = \angle DCA
∠
CB
D
=
∠
D
C
A
,
A
B
=
15
AB = 15
A
B
=
15
,
C
D
=
8
CD = 8
C
D
=
8
. Show that
A
B
C
D
ABCD
A
BC
D
is cyclic and find the distance between its circumcenter and the intersection point of its diagonals.
2
2
Hide problems
3 points split a segment into 4 equal parts wanted
Let
A
B
C
ABC
A
BC
be a triangle with centroid
T
T
T
. Denote by
M
M
M
the midpoint of
B
C
BC
BC
. Let
D
D
D
be a point on the ray opposite to the ray
B
A
BA
B
A
such that
A
B
=
B
D
AB = BD
A
B
=
B
D
. Similarly, let
E
E
E
be a point on the ray opposite to the ray
C
A
CA
C
A
such that
A
C
=
C
E
AC = CE
A
C
=
CE
. The segments
T
D
T D
T
D
and
T
E
T E
TE
intersect the side
B
C
BC
BC
in
P
P
P
and
Q
Q
Q
, respectively. Show that the points
P
,
Q
P, Q
P
,
Q
and
M
M
M
split the segment
B
C
BC
BC
into four parts of equal length.
sick rooks in a square chess board
The chess piece sick rook can move along rows and columns as a regular rook, but at most by
2
2
2
fields. We can place sick rooks on a square board in such a way that no two of them attack each other and no field is attacked by more than one sick rook. a) Prove that on
30
×
30
30\times 30
30
×
30
board, we cannot place more than
100
100
100
sick rooks. b) Find the maximum number of sick rooks which can be placed on
8
×
8
8\times 8
8
×
8
board. c) Prove that on
32
×
32
32\times 32
32
×
32
board, we cannot place more than
120
120
120
sick rooks.
1
2
Hide problems
a+b and a^2+b^2 are integers , then rational a,b are integers
Rational numbers
a
,
b
a, b
a
,
b
are such that
a
+
b
a+b
a
+
b
and
a
2
+
b
2
a^2+b^2
a
2
+
b
2
are integers. Prove that
a
,
b
a, b
a
,
b
are integers.
radical diophantine \sqrt{a+2\sqrt{b}}=\sqrt{a-2\sqrt{b}}+\sqrt{b}
Find all pairs of positive integers
a
,
b
a, b
a
,
b
such that
a
+
2
b
=
a
−
2
b
+
b
\sqrt{a+2\sqrt{b}}=\sqrt{a-2\sqrt{b}}+\sqrt{b}
a
+
2
b
=
a
−
2
b
+
b
.