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Problems
Contests
International Contests
International Zhautykov Olympiad
2019 International Zhautykov OIympiad
2019 International Zhautykov OIympiad
Part of
International Zhautykov Olympiad
Subcontests
(6)
6
1
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Polynomial (Izho)
We define two types of operation on polynomial of third degree: a) switch places of the coefficients of polynomial(including zero coefficients), ex:
x
3
+
x
2
+
3
x
−
2
x^3+x^2+3x-2
x
3
+
x
2
+
3
x
−
2
=>
−
2
x
3
+
3
x
2
+
x
+
1
-2x^3+3x^2+x+1
−
2
x
3
+
3
x
2
+
x
+
1
b) replace the polynomial
P
(
x
)
P(x)
P
(
x
)
with
P
(
x
+
1
)
P(x+1)
P
(
x
+
1
)
If limitless amount of operations is allowed, is it possible from
x
3
−
2
x^3-2
x
3
−
2
to get
x
3
−
3
x
2
+
3
x
−
3
x^3-3x^2+3x-3
x
3
−
3
x
2
+
3
x
−
3
?
5
1
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Period of function (Izho)
Natural number
n
>
1
n>1
n
>
1
is given. Let
I
I
I
be a set of integers that are relatively prime to
n
n
n
. Define the function
f
:
I
=
>
N
f:I=>N
f
:
I
=>
N
. We call a function
k
−
p
e
r
i
o
d
i
c
k-periodic
k
−
p
er
i
o
d
i
c
if for any
a
,
b
a,b
a
,
b
,
f
(
a
)
=
f
(
b
)
f(a)=f(b)
f
(
a
)
=
f
(
b
)
whenever
k
∣
a
−
b
k|a-b
k
∣
a
−
b
. We know that
f
f
f
is
n
−
p
e
r
i
o
d
i
c
n-periodic
n
−
p
er
i
o
d
i
c
. Prove that minimal period of
f
f
f
divides all other periods. Example: if
n
=
6
n=6
n
=
6
and
f
(
1
)
=
f
(
5
)
f(1)=f(5)
f
(
1
)
=
f
(
5
)
then minimal period is 1, if
f
(
1
)
f(1)
f
(
1
)
is not equal to
f
(
5
)
f(5)
f
(
5
)
then minimal period is 3.
4
1
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Another geometry(Izho)
Triangle
A
B
C
ABC
A
BC
with
A
C
=
B
C
AC=BC
A
C
=
BC
given and point
D
D
D
is chosen on the side
A
C
AC
A
C
.
S
1
S1
S
1
is a circle that touches
A
D
AD
A
D
and extensions of
A
B
AB
A
B
and
B
D
BD
B
D
with radius
R
R
R
and center
O
1
O_1
O
1
.
S
2
S2
S
2
is a circle that touches
C
D
CD
C
D
and extensions of
B
C
BC
BC
and
B
D
BD
B
D
with radius
2
R
2R
2
R
and center
O
2
O_2
O
2
. Let
F
F
F
be intersection of the extension of
A
B
AB
A
B
and tangent at
O
2
O_2
O
2
to circumference of
B
O
1
O
2
BO_1O_2
B
O
1
O
2
. Prove that
F
O
1
=
O
1
O
2
FO_1=O_1O_2
F
O
1
=
O
1
O
2
.
3
1
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Geometry from Izho(P3)
Triangle
A
B
C
ABC
A
BC
is given. The median
C
M
CM
CM
intersects the circumference of
A
B
C
ABC
A
BC
in
N
N
N
.
P
P
P
and
Q
Q
Q
are chosen on the rays
C
A
CA
C
A
and
C
B
CB
CB
respectively, such that
P
M
PM
PM
is parallel to
B
N
BN
BN
and
Q
M
QM
QM
is parallel to
A
N
AN
A
N
. Points
X
X
X
and
Y
Y
Y
are chosen on the segments
P
M
PM
PM
and
Q
M
QM
QM
respectively, such that both
P
Y
PY
P
Y
and
Q
X
QX
QX
touch the circumference of
A
B
C
ABC
A
BC
. Let
Z
Z
Z
be intersection of
P
Y
PY
P
Y
and
Q
X
QX
QX
. Prove that, the quadrilateral
M
X
Z
Y
MXZY
MXZ
Y
is circumscribed.
2
1
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IZHO 2019 P2
Find the biggest real number
C
C
C
, such that for every different positive real numbers
a
1
,
a
2
.
.
.
a
2019
a_1,a_2...a_{2019}
a
1
,
a
2
...
a
2019
that satisfy inequality :
a
1
∣
a
2
−
a
3
∣
+
a
2
∣
a
3
−
a
4
∣
+
.
.
.
+
a
2019
∣
a
1
−
a
2
∣
>
C
\frac{a_1}{|a_2-a_3|} + \frac{a_2}{|a_3-a_4|} + ... + \frac{a_{2019}}{|a_1-a_2|} > C
∣
a
2
−
a
3
∣
a
1
+
∣
a
3
−
a
4
∣
a
2
+
...
+
∣
a
1
−
a
2
∣
a
2019
>
C
1
1
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Comb problem from IZHO 2019
Prove that there exist at least
100
!
100!
100
!
ways to write
100
!
100!
100
!
as sum of elements of set {
1
!
,
2
!
,
3
!
.
.
.
99
!
1!,2!,3!...99!
1
!
,
2
!
,
3
!
...99
!
} (each number in sum can be two or more times)