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Problems
Contests
International Contests
International Zhautykov Olympiad
2017 International Zhautykov Olympiad
2017 International Zhautykov Olympiad
Part of
International Zhautykov Olympiad
Subcontests
(3)
3
2
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Tilings with domino figures
Rectangle on a checked paper with length of a unit square side being
1
1
1
Is divided into domino figures( two unit square sharing a common edge). Prove that you colour all corners of squares on the edge of rectangle and inside rectangle with
3
3
3
colours such that for any two corners with distance
1
1
1
the following conditions hold: they are coloured in different colour if the line connecting the two corners is on the border of two domino figures and coloured in same colour if the line connecting the two corners is inside a domino figure.
Tetrahedron inequality
Let
A
B
C
D
ABCD
A
BC
D
be the regular tetrahedron, and
M
,
N
M, N
M
,
N
points in space. Prove that:
A
M
⋅
A
N
+
B
M
⋅
B
N
+
C
M
⋅
C
N
≥
D
M
⋅
D
N
AM \cdot AN + BM \cdot BN + CM \cdot CN \geq DM \cdot DN
A
M
⋅
A
N
+
BM
⋅
BN
+
CM
⋅
CN
≥
D
M
⋅
D
N
1
2
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Concyclic points
Let
A
B
C
ABC
A
BC
be a non-isosceles triangle with circumcircle
ω
\omega
ω
and let
H
,
M
H, M
H
,
M
be orthocenter and midpoint of
A
B
AB
A
B
respectively. Let
P
,
Q
P,Q
P
,
Q
be points on the arc
A
B
AB
A
B
of
ω
\omega
ω
not containing
C
C
C
such that
∠
A
C
P
=
∠
B
C
Q
<
∠
A
C
Q
\angle ACP=\angle BCQ < \angle ACQ
∠
A
CP
=
∠
BCQ
<
∠
A
CQ
.Let
R
,
S
R,S
R
,
S
be the foot of altitudes from
H
H
H
to
C
Q
,
C
P
CQ,CP
CQ
,
CP
respectively. Prove that thé points
P
,
Q
,
R
,
S
P,Q,R,S
P
,
Q
,
R
,
S
are concyclic and
M
M
M
is the center of this circle.
Sequnce of positive integers
Let
(
a
n
)
(a_n)
(
a
n
)
be sequnce of positive integers such that first
k
k
k
members
a
1
,
a
2
,
.
.
.
,
a
k
a_1,a_2,...,a_k
a
1
,
a
2
,
...
,
a
k
are distinct positive integers, and for each
n
>
k
n>k
n
>
k
, number
a
n
a_n
a
n
is the smallest positive integer that can't be represented as a sum of several (possibly one) of the numbers
a
1
,
a
2
,
.
.
.
,
a
n
−
1
a_1,a_2,...,a_{n-1}
a
1
,
a
2
,
...
,
a
n
−
1
. Prove that
a
n
=
2
a
n
−
1
a_n=2a_{n-1}
a
n
=
2
a
n
−
1
for all sufficently large
n
n
n
.
2
2
Hide problems
Sum of distinct prime divisors
For each positive integer
k
k
k
denote
C
(
k
)
C(k)
C
(
k
)
to be sum of its distinct prime divisors. For example
C
(
1
)
=
0
,
C
(
2
)
=
2
,
C
(
45
)
=
8
C(1)=0,C(2)=2,C(45)=8
C
(
1
)
=
0
,
C
(
2
)
=
2
,
C
(
45
)
=
8
. Find all positive integers
n
n
n
for which
C
(
2
n
+
1
)
=
C
(
n
)
C(2^n+1)=C(n)
C
(
2
n
+
1
)
=
C
(
n
)
.
IZHO 2017 Functional equations
Find all functions
f
:
R
→
R
f:R \rightarrow R
f
:
R
→
R
such that
(
x
+
y
2
)
f
(
y
f
(
x
)
)
=
x
y
f
(
y
2
+
f
(
x
)
)
(x+y^2)f(yf(x))=xyf(y^2+f(x))
(
x
+
y
2
)
f
(
y
f
(
x
))
=
x
y
f
(
y
2
+
f
(
x
))
, where
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R