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Problems
Contests
International Contests
International Zhautykov Olympiad
2013 International Zhautykov Olympiad
2013 International Zhautykov Olympiad
Part of
International Zhautykov Olympiad
Subcontests
(3)
3
2
Hide problems
The big inequality with condition abcd = 1
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
, and
d
d
d
be positive real numbers such that
a
b
c
d
=
1
abcd = 1
ab
c
d
=
1
. Prove that
(
a
−
1
)
(
c
+
1
)
1
+
b
c
+
c
+
(
b
−
1
)
(
d
+
1
)
1
+
c
d
+
d
+
(
c
−
1
)
(
a
+
1
)
1
+
d
a
+
a
+
(
d
−
1
)
(
b
+
1
)
1
+
a
b
+
b
≥
0.
\frac{(a-1)(c+1)}{1+bc+c} + \frac{(b-1)(d+1)}{1+cd+d} + \frac{(c-1)(a+1)}{1+da+a} + \frac{(d-1)(b+1)}{1+ab+b} \geq 0.
1
+
b
c
+
c
(
a
−
1
)
(
c
+
1
)
+
1
+
c
d
+
d
(
b
−
1
)
(
d
+
1
)
+
1
+
d
a
+
a
(
c
−
1
)
(
a
+
1
)
+
1
+
ab
+
b
(
d
−
1
)
(
b
+
1
)
≥
0.
Proposed by Orif Ibrogimov, Uzbekistan.
Blocks in a 10 X 10 table
A
10
×
10
10 \times 10
10
×
10
table consists of
100
100
100
unit cells. A block is a
2
×
2
2 \times 2
2
×
2
square consisting of
4
4
4
unit cells of the table. A set
C
C
C
of
n
n
n
blocks covers the table (i.e. each cell of the table is covered by some block of
C
C
C
) but no
n
−
1
n -1
n
−
1
blocks of
C
C
C
cover the table. Find the largest possible value of
n
n
n
.
2
2
Hide problems
All n which divide one of the two numbers
Find all odd positive integers
n
>
1
n>1
n
>
1
such that there is a permutation
a
1
,
a
2
,
a
3
,
…
,
a
n
a_1, a_2, a_3, \ldots, a_n
a
1
,
a
2
,
a
3
,
…
,
a
n
of the numbers
1
,
2
,
3
,
…
,
n
1, 2,3, \ldots, n
1
,
2
,
3
,
…
,
n
where
n
n
n
divides one of the numbers
a
k
2
−
a
k
+
1
−
1
a_k^2 - a_{k+1} - 1
a
k
2
−
a
k
+
1
−
1
and
a
k
2
−
a
k
+
1
+
1
a_k^2 - a_{k+1} + 1
a
k
2
−
a
k
+
1
+
1
for each
k
k
k
,
1
≤
k
≤
n
1 \leq k \leq n
1
≤
k
≤
n
(we assume
a
n
+
1
=
a
1
a_{n+1}=a_1
a
n
+
1
=
a
1
).
The sum of lengths does not exceed the perimeter
Given convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
with
A
B
∥
D
E
AB \parallel DE
A
B
∥
D
E
,
B
C
∥
E
F
BC \parallel EF
BC
∥
EF
, and
C
D
∥
F
A
CD \parallel FA
C
D
∥
F
A
. The distance between the lines
A
B
AB
A
B
and
D
E
DE
D
E
is equal to the distance between the lines
B
C
BC
BC
and
E
F
EF
EF
and to the distance between the lines
C
D
CD
C
D
and
F
A
FA
F
A
. Prove that the sum
A
D
+
B
E
+
C
F
AD+BE+CF
A
D
+
BE
+
CF
does not exceed the perimeter of hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
.
1
2
Hide problems
Prove that the line O1O2 passes through the point N
Given a trapezoid
A
B
C
D
ABCD
A
BC
D
(
A
D
∥
B
C
AD \parallel BC
A
D
∥
BC
) with
∠
A
B
C
>
9
0
∘
\angle ABC > 90^\circ
∠
A
BC
>
9
0
∘
. Point
M
M
M
is chosen on the lateral side
A
B
AB
A
B
. Let
O
1
O_1
O
1
and
O
2
O_2
O
2
be the circumcenters of the triangles
M
A
D
MAD
M
A
D
and
M
B
C
MBC
MBC
, respectively. The circumcircles of the triangles
M
O
1
D
MO_1D
M
O
1
D
and
M
O
2
C
MO_2C
M
O
2
C
meet again at the point
N
N
N
. Prove that the line
O
1
O
2
O_1O_2
O
1
O
2
passes through the point
N
N
N
.
The equation p(x) = 1/n has no rational roots
A quadratic trinomial
p
(
x
)
p(x)
p
(
x
)
with real coefficients is given. Prove that there is a positive integer
n
n
n
such that the equation
p
(
x
)
=
1
n
p(x) = \frac{1}{n}
p
(
x
)
=
n
1
has no rational roots.