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Problems
Contests
International Contests
International Zhautykov Olympiad
2022 International Zhautykov Olympiad
2022 International Zhautykov Olympiad
Part of
International Zhautykov Olympiad
Subcontests
(6)
6
1
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Bounded Sequences
Do there exist two bounded sequences
a
1
,
a
2
,
…
a_1, a_2,\ldots
a
1
,
a
2
,
…
and
b
1
,
b
2
,
…
b_1, b_2,\ldots
b
1
,
b
2
,
…
such that for each positive integers
n
n
n
and
m
>
n
m>n
m
>
n
at least one of the two inequalities
∣
a
m
−
a
n
∣
>
1
/
n
,
|a_m-a_n|>1/\sqrt{n},
∣
a
m
−
a
n
∣
>
1/
n
,
and
∣
b
m
−
b
n
∣
>
1
/
n
|b_m-b_n|>1/\sqrt{n}
∣
b
m
−
b
n
∣
>
1/
n
holds?
4
1
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Length of segments
In triangle
A
B
C
ABC
A
BC
, a point
M
M
M
is the midpoint of
A
B
AB
A
B
, and a point
I
I
I
is the incentre. Point
A
1
A_1
A
1
is the reflection of
A
A
A
in
B
I
BI
B
I
, and
B
1
B_1
B
1
is the reflection of
B
B
B
in
A
I
AI
A
I
. Let
N
N
N
be the midpoint of
A
1
B
1
A_1B_1
A
1
B
1
. Prove that
I
N
>
I
M
IN > IM
I
N
>
I
M
.
5
1
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Sums Of Polynomials
A polynomial
f
(
x
)
f(x)
f
(
x
)
with real coefficients of degree greater than
1
1
1
is given. Prove that there are infinitely many positive integers which cannot be represented in the form
f
(
n
+
1
)
+
f
(
n
+
2
)
+
⋯
+
f
(
n
+
k
)
f(n+1)+f(n+2)+\cdots+f(n+k)
f
(
n
+
1
)
+
f
(
n
+
2
)
+
⋯
+
f
(
n
+
k
)
where
n
n
n
and
k
k
k
are positive integers.
3
1
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Equal angles in a parallelogram
In parallelogram
A
B
C
D
ABCD
A
BC
D
with acute angle
A
A
A
a point
N
N
N
is chosen on the segment
A
D
AD
A
D
, and a point
M
M
M
on the segment
C
N
CN
CN
so that
A
B
=
B
M
=
C
M
AB = BM = CM
A
B
=
BM
=
CM
. Point
K
K
K
is the reflection of
N
N
N
in line
M
D
MD
M
D
. The line
M
K
MK
M
K
meets the segment
A
D
AD
A
D
at point
L
L
L
. Let
P
P
P
be the common point of the circumcircles of
A
M
D
AMD
A
M
D
and
C
N
K
CNK
CN
K
such that
A
A
A
and
P
P
P
share the same side of the line
M
K
MK
M
K
. Prove that
∠
C
P
M
=
∠
D
P
L
\angle CPM = \angle DPL
∠
CPM
=
∠
D
P
L
.
2
1
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Colouring of a 2-tree
A ten-level
2
2
2
-tree is drawn in the plane: a vertex
A
1
A_1
A
1
is marked, it is connected by segments with two vertices
B
1
B_1
B
1
and
B
2
B_2
B
2
, each of
B
1
B_1
B
1
and
B
2
B_2
B
2
is connected by segments with two of the four vertices
C
1
,
C
2
,
C
3
,
C
4
C_1, C_2, C_3, C_4
C
1
,
C
2
,
C
3
,
C
4
(each
C
i
C_i
C
i
is connected with one
B
j
B_j
B
j
exactly); and so on, up to
512
512
512
vertices
J
1
,
…
,
J
512
J_1, \ldots, J_{512}
J
1
,
…
,
J
512
. Each of the vertices
J
1
,
…
,
J
512
J_1, \ldots, J_{512}
J
1
,
…
,
J
512
is coloured blue or golden. Consider all permutations
f
f
f
of the vertices of this tree, such that (i) if
X
X
X
and
Y
Y
Y
are connected with a segment, then so are
f
(
X
)
f(X)
f
(
X
)
and
f
(
Y
)
f(Y)
f
(
Y
)
, and (ii) if
X
X
X
is coloured, then
f
(
X
)
f(X)
f
(
X
)
has the same colour. Find the maximum
M
M
M
such that there are at least
M
M
M
permutations with these properties, regardless of the colouring.
1
1
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Polynomial Identity
Non-zero polynomials
P
(
x
)
P(x)
P
(
x
)
,
Q
(
x
)
Q(x)
Q
(
x
)
, and
R
(
x
)
R(x)
R
(
x
)
with real coefficients satisfy the identities
P
(
x
)
+
Q
(
x
)
+
R
(
x
)
=
P
(
Q
(
x
)
)
+
Q
(
R
(
x
)
)
+
R
(
P
(
x
)
)
=
0.
P(x) + Q(x) + R(x) = P(Q(x)) + Q(R(x)) + R(P(x)) = 0.
P
(
x
)
+
Q
(
x
)
+
R
(
x
)
=
P
(
Q
(
x
))
+
Q
(
R
(
x
))
+
R
(
P
(
x
))
=
0.
Prove that the degrees of the three polynomials are all even.