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Problems
Contests
International Contests
International Zhautykov Olympiad
2015 International Zhautykov Olympiad
2015 International Zhautykov Olympiad
Part of
International Zhautykov Olympiad
Subcontests
(3)
2
2
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Nice geometry problem
Inside the triangle
A
B
C
ABC
A
BC
a point
M
M
M
is given. The line
B
M
BM
BM
meets the side
A
C
AC
A
C
at
N
N
N
. The point
K
K
K
is symmetrical to
M
M
M
with respect to
A
C
AC
A
C
. The line
B
K
BK
B
K
meets
A
C
AC
A
C
at
P
P
P
. If
∠
A
M
P
=
∠
C
M
N
\angle AMP = \angle CMN
∠
A
MP
=
∠
CMN
, prove that
∠
A
B
P
=
∠
C
B
N
\angle ABP=\angle CBN
∠
A
BP
=
∠
CBN
.
combinatorics with some sets
Let
A
n
A_n
A
n
be the set of partitions of the sequence
1
,
2
,
.
.
.
,
n
1,2,..., n
1
,
2
,
...
,
n
into several subsequences such that every two neighbouring terms of each subsequence have different parity,and
B
n
B_n
B
n
the set of partitions of the sequence
1
,
2
,
.
.
.
,
n
1,2,..., n
1
,
2
,
...
,
n
into several subsequences such that all the terms of each subsequence have the same parity ( for example,the partition
(
1
,
4
,
5
,
8
)
,
(
2
,
3
)
,
(
6
,
9
)
,
(
7
)
{(1,4,5,8),(2,3),(6,9),(7)}
(
1
,
4
,
5
,
8
)
,
(
2
,
3
)
,
(
6
,
9
)
,
(
7
)
is an element of
A
9
A_9
A
9
,and the partition
(
1
,
3
,
5
)
,
(
2
,
4
)
,
(
6
)
{(1,3,5),(2,4),(6)}
(
1
,
3
,
5
)
,
(
2
,
4
)
,
(
6
)
is an element of
B
6
B_6
B
6
). Prove that for every positive integer
n
n
n
the sets
A
n
A_n
A
n
and
B
n
+
1
B_{n+1}
B
n
+
1
contain the same number of elements.
1
2
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triangle with area n
Each point with integral coordinates in the plane is coloured white or blue. Prove that one can choose a colour so that for every positive integer
n
n
n
there exists a triangle of area
n
n
n
having its vertices of the chosen colour.
determine the maximum n
Determine the maximum integer
n
n
n
such that for each positive integer
k
≤
n
2
k \le \frac{n}{2}
k
≤
2
n
there are two positive divisors of
n
n
n
with difference
k
k
k
.
3
2
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Hard functional equation
Find all functions
f
:
R
→
R
f\colon \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
x
3
+
y
3
+
x
y
)
=
x
2
f
(
x
)
+
y
2
f
(
y
)
+
f
(
x
y
)
f(x^3+y^3+xy)=x^2f(x)+y^2f(y)+f(xy)
f
(
x
3
+
y
3
+
x
y
)
=
x
2
f
(
x
)
+
y
2
f
(
y
)
+
f
(
x
y
)
, for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
.
pentagon inequality
The area of a convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
is
S
S
S
, and the circumradii of the triangles
A
B
C
ABC
A
BC
,
B
C
D
BCD
BC
D
,
C
D
E
CDE
C
D
E
,
D
E
A
DEA
D
E
A
,
E
A
B
EAB
E
A
B
are
R
1
R_1
R
1
,
R
2
R_2
R
2
,
R
3
R_3
R
3
,
R
4
R_4
R
4
,
R
5
R_5
R
5
. Prove the inequality
R
1
4
+
R
2
4
+
R
3
4
+
R
4
4
+
R
5
4
≥
4
5
sin
2
10
8
∘
S
2
.
R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2.
R
1
4
+
R
2
4
+
R
3
4
+
R
4
4
+
R
5
4
≥
5
sin
2
10
8
∘
4
S
2
.