MathDB
Problems
Contests
International Contests
International Zhautykov Olympiad
2016 International Zhautykov Olympiad
2016 International Zhautykov Olympiad
Part of
International Zhautykov Olympiad
Subcontests
(3)
3
2
Hide problems
izho 2016 p3
There are
60
60
60
towns in
G
r
a
p
h
l
a
n
d
Graphland
G
r
a
p
h
l
an
d
every two countries of which are connected by only a directed way. Prove that we can color four towns to red and four towns to green such that every way between green and red towns are directed from red to green
Izho 2016 p6
We call a positive integer
q
q
q
a convenient denominator for a real number
α
\alpha
α
if
∣
α
−
p
q
∣
<
1
10
q
\displaystyle |\alpha - \dfrac{p}{q}|<\dfrac{1}{10q}
∣
α
−
q
p
∣
<
10
q
1
for some integer
p
p
p
. Prove that if two irrational numbers
α
\alpha
α
and
β
\beta
β
have the same set of convenient denominators then either
α
+
β
\alpha+\beta
α
+
β
or
α
−
β
\alpha- \beta
α
−
β
is an integer.
2
2
Hide problems
Number Theory
a
1
,
a
2
,
.
.
.
,
a
100
a_1,a_2,...,a_{100}
a
1
,
a
2
,
...
,
a
100
are permutation of
1
,
2
,
.
.
.
,
100
1,2,...,100
1
,
2
,
...
,
100
.
S
1
=
a
1
,
S
2
=
a
1
+
a
2
,
.
.
.
,
S
100
=
a
1
+
a
2
+
.
.
.
+
a
100
S_1=a_1, S_2=a_1+a_2,...,S_{100}=a_1+a_2+...+a_{100}
S
1
=
a
1
,
S
2
=
a
1
+
a
2
,
...
,
S
100
=
a
1
+
a
2
+
...
+
a
100
Find the maximum number of perfect squares from
S
i
S_i
S
i
Izho 2016P5
A convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is given such that
A
B
∣
∣
D
E
AB||DE
A
B
∣∣
D
E
,
B
C
∣
∣
E
F
BC||EF
BC
∣∣
EF
,
C
D
∣
∣
F
A
CD||FA
C
D
∣∣
F
A
. The point
M
,
N
,
K
M, N, K
M
,
N
,
K
are common points of the lines
B
D
BD
B
D
and
A
E
AE
A
E
,
A
C
AC
A
C
and
D
F
DF
D
F
,
C
E
CE
CE
and
B
F
BF
BF
respectively. Prove that perpendiculars drawn from
M
,
N
,
K
M, N, K
M
,
N
,
K
to lines
A
B
,
C
D
,
E
F
AB, CD, EF
A
B
,
C
D
,
EF
respectively concurrent.
1
2
Hide problems
Geometry
A quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle with center
O
O
O
. It's diagonals meet at
M
M
M
.The circumcircle of
A
B
M
ABM
A
BM
intersects the sides
A
D
AD
A
D
and
B
C
BC
BC
at
N
N
N
and
K
K
K
respectively. Prove that areas of
N
O
M
D
NOMD
NOM
D
and
K
O
M
C
KOMC
K
OMC
are equal.
Functional inequality
Find all
k
>
0
k>0
k
>
0
for which a strictly decreasing function
g
:
(
0
;
+
∞
)
→
(
0
;
+
∞
)
g:(0;+\infty)\to(0;+\infty)
g
:
(
0
;
+
∞
)
→
(
0
;
+
∞
)
exists such that
g
(
x
)
≥
k
g
(
x
+
g
(
x
)
)
g(x)\geq kg(x+g(x))
g
(
x
)
≥
k
g
(
x
+
g
(
x
))
for all positive
x
x
x
.