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International Contests
Czech-Polish-Slovak Junior Match
2022 Czech-Polish-Slovak Junior Match
2022 Czech-Polish-Slovak Junior Match
Part of
Czech-Polish-Slovak Junior Match
Subcontests
(6)
3
2
Hide problems
convex pentagon ABCD fits in a circle with radius 2/3 AD
Given is a convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
in which
∠
A
=
6
0
o
\angle A = 60^o
∠
A
=
6
0
o
,
∠
B
=
10
0
o
\angle B = 100^o
∠
B
=
10
0
o
,
∠
C
=
14
0
o
\angle C = 140^o
∠
C
=
14
0
o
. Show that this pentagon can be placed in a circle with a radius of
2
3
A
D
\frac23 AD
3
2
A
D
.
PQ//BC wanted, FB = BD, DC = CE
The points
D
,
E
,
F
D, E, F
D
,
E
,
F
lie respectively on the sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
of the triangle ABC such that
F
B
=
B
D
F B = BD
FB
=
B
D
,
D
C
=
C
E
DC = CE
D
C
=
CE
, and the lines
E
F
EF
EF
and
B
C
BC
BC
are parallel. Tangent to the circumscribed circle of triangle
D
E
F
DEF
D
EF
at point
F
F
F
intersects line
A
D
AD
A
D
at point
P
P
P
. Perpendicular bisector of segment
E
F
EF
EF
intersects the segment
A
C
AC
A
C
at
Q
Q
Q
. Prove that the lines
P
Q
P Q
PQ
and
B
C
BC
BC
are parallel.
6
1
Hide problems
bw coloring of 6x6 table
Find all integers
n
≥
4
n \ge 4
n
≥
4
with the following property: Each field of the
n
×
n
n \times n
n
×
n
table can be painted white or black in such a way that each square of this table had the same color as exactly the two adjacent squares. (Squares are adjacent if they have exactly one side in common.) How many different colorings of the
6
×
6
6 \times 6
6
×
6
table fields meet the above conditions?
5
2
Hide problems
integer divisible by n + 1, n + 2, ..., n + 9 also by n+10
An integer
n
≥
1
n\ge1
n
≥
1
is good if the following property is satisfied: If a positive integer is divisible by each of the nine numbers
n
+
1
,
n
+
2
,
.
.
.
,
n
+
9
n + 1, n + 2, ..., n + 9
n
+
1
,
n
+
2
,
...
,
n
+
9
, this is also divisible by
n
+
10
n + 10
n
+
10
. How many good integers are
n
≥
1
n\ge 1
n
≥
1
?
computational geo with regular nonagon
Given a regular nonagon
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
A
9
A_1A_2A_3A_4A_5A_6A_7A_8A_9
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
A
9
with side length
1
1
1
. Diagonals
A
3
A
7
A_3A_7
A
3
A
7
and
A
4
A
8
A_4A_8
A
4
A
8
intersect at point
P
P
P
. Find the length of segment
P
A
1
P A_1
P
A
1
.
4
2
Hide problems
a/b - 1/2ab > \sqrt2 for pos.integers with a/b > \sqrt2
Let
a
a
a
and
b
b
b
be positive integers with the property that
a
b
>
2
\frac{a}{b} > \sqrt2
b
a
>
2
. Prove that
a
b
−
1
2
a
b
>
2
\frac{a}{b} - \frac{1}{2ab} > \sqrt2
b
a
−
2
ab
1
>
2
diophantine systema, a + b = c , a^2 + b^3 = c^2
Find all triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of integers that satisfy the equations
a
+
b
=
c
a + b = c
a
+
b
=
c
and
a
2
+
b
3
=
c
2
a^2 + b^3 = c^2
a
2
+
b
3
=
c
2
2
2
Hide problems
diophantine system x^2 = yz + 1 , y^2 = zx + 1, z^2 = xy + 1
Solve the following system of equations in integer numbers:
{
x
2
=
y
z
+
1
y
2
=
z
x
+
1
z
2
=
x
y
+
1
\begin{cases} x^2 = yz + 1 \\ y^2 = zx + 1 \\ z^2 = xy + 1 \end{cases}
⎩
⎨
⎧
x
2
=
yz
+
1
y
2
=
z
x
+
1
z
2
=
x
y
+
1
1st number divisible by 22, 2022 => 2020222 => 2020220222 => ...,
The number
2022
2022
2022
is written on the board. In each step, we replace one of the
2
2
2
digits with the number
2022
2022
2022
. For example
2022
⇒
2020222
⇒
2020220222
⇒
.
.
.
2022 \Rightarrow 2020222 \Rightarrow 2020220222 \Rightarrow ...
2022
⇒
2020222
⇒
2020220222
⇒
...
After how many steps can a number divisible by
22
22
22
be written on the board? Specify all options.
1
2
Hide problems
min different values of sums of 2 reals out of n
Let
n
≥
3
n\ge 3
n
≥
3
. Suppose
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ... , a_n
a
1
,
a
2
,
...
,
a
n
are
n
n
n
distinct in pairs real numbers. In terms of
n
n
n
, find the smallest possible number of different assumed values by the following
n
n
n
numbers:
a
1
+
a
2
,
a
2
+
a
3
,
.
.
.
,
a
n
−
1
+
a
n
,
a
n
+
a
1
a_1 + a_2, a_2 + a_3,..., a_{n- 1} + a_n, a_n + a_1
a
1
+
a
2
,
a
2
+
a
3
,
...
,
a
n
−
1
+
a
n
,
a
n
+
a
1
max of ab+bc+ 2ac for a,b,c>=0 with a+b+c=1
Determine the largest possible value of the expression
a
b
+
b
c
+
2
a
c
ab+bc+ 2ac
ab
+
b
c
+
2
a
c
for non-negative real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
whose sum is
1
1
1
.