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Contests
International Contests
International Zhautykov Olympiad
2024 International Zhautykov Olympiad
2024 International Zhautykov Olympiad
Part of
International Zhautykov Olympiad
Subcontests
(6)
4
1
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Uninspiring algebra
Ten distinct positive real numbers are given and the sum of each pair is written (So 45 sums). Between these sums there are 5 equal numbers. If we calculate product of each pair, find the biggest number
k
k
k
such that there may be
k
k
k
equal numbers between them.
5
1
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Dominoes and 3x1 stripes
We are given
m
×
n
m\times n
m
×
n
table tiled with
3
×
1
3\times 1
3
×
1
stripes and we are given that
6
∣
m
n
6 | mn
6∣
mn
. Prove that there exists a tiling of the table with
2
×
1
2\times 1
2
×
1
dominoes such that each of these stripes contains one whole domino.
6
1
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Maximal angles between medians and sides
Let
G
G
G
be the centroid of triangle
A
B
C
ABC
A
BC
. Find the biggest
α
\alpha
α
such that there exists a triangle for which there are at least three angles among
∠
G
A
B
,
∠
G
A
C
,
∠
G
B
A
,
∠
G
B
C
,
∠
G
C
A
,
∠
G
C
B
\angle GAB, \angle GAC, \angle GBA, \angle GBC, \angle GCA, \angle GCB
∠
G
A
B
,
∠
G
A
C
,
∠
GB
A
,
∠
GBC
,
∠
GC
A
,
∠
GCB
which are
≥
α
\geq \alpha
≥
α
.
3
1
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IZhO 2024, P3
Positive integer
d
d
d
is not perfect square. For each positive integer
n
n
n
, let
s
(
n
)
s(n)
s
(
n
)
denote the number of digits
1
1
1
among the first
n
n
n
digits in the binary representation of
d
\sqrt{d}
d
(including the digits before the point). Prove that there exists an integer
A
A
A
such that
s
(
n
)
>
2
n
−
2
s(n)>\sqrt{2n}-2
s
(
n
)
>
2
n
−
2
for all integers
n
≥
A
n\ge A
n
≥
A
2
1
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IZhO 2024, P2
Circles
Ω
\Omega
Ω
and
Γ
\Gamma
Γ
meet at points
A
A
A
and
B
B
B
. The line containing their centres intersects
Ω
\Omega
Ω
and
Γ
\Gamma
Γ
at point
P
P
P
and
Q
Q
Q
, respectively, such that these points lie on same side of the line
A
B
AB
A
B
and point
Q
Q
Q
is closer to
A
B
AB
A
B
than point
P
P
P
. The circle
δ
\delta
δ
lies on the same side of the line
A
B
AB
A
B
as
P
P
P
and
Q
Q
Q
, touches the segment
A
B
AB
A
B
at point
D
D
D
and touches
Γ
\Gamma
Γ
at point
T
T
T
. The line
P
D
PD
P
D
meets
δ
\delta
δ
and
Ω
\Omega
Ω
again at points
K
K
K
and
L
L
L
, respectively. Prove that
∠
Q
T
K
=
∠
D
T
L
\angle QTK=\angle DTL
∠
QT
K
=
∠
D
T
L
1
1
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IZhO 2024, P1
In an alphabet of
n
n
n
letters, is
s
y
l
l
a
b
l
e
syllable
sy
ll
ab
l
e
is any ordered pair of two (not necessarily distinct) letters. Some syllables are considered
i
n
d
e
c
e
n
t
indecent
in
d
ece
n
t
. A
w
o
r
d
word
w
or
d
is any sequence, finite or infinite, of letters, that does not contain indecent syllables. Find the least possible number of indecent syllables for which infinite words do not exist.