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Problems
Contests
International Contests
International Zhautykov Olympiad
2009 International Zhautykov Olympiad
2009 International Zhautykov Olympiad
Part of
International Zhautykov Olympiad
Subcontests
(3)
3
2
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An Outstanding Inequality in an arbitrary hexagon, ZIMO - 09
For a convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
with an area
S
S
S
, prove that: AC\cdot(BD\plus{}BF\minus{}DF)\plus{}CE\cdot(BD\plus{}DF\minus{}BF)\plus{}AE\cdot(BF\plus{}DF\minus{}BD)\geq 2\sqrt{3}S
17*17 table with some cells colored in black, ZIMO - 09
In a checked
17
×
17
17\times 17
17
×
17
table,
n
n
n
squares are colored in black. We call a line any of rows, columns, or any of two diagonals of the table. In one step, if at least
6
6
6
of the squares in some line are black, then one can paint all the squares of this line in black. Find the minimal value of
n
n
n
such that for some initial arrangement of
n
n
n
black squares one can paint all squares of the table in black in some steps.
2
2
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Beautiful functional equation-inequality, ZIMO-09
Find all real
a
a
a
, such that there exist a function
f
:
R
→
R
f: \mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
satisfying the following inequality: x\plus{}af(y)\leq y\plus{}f(f(x)) for all
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
Quadrilateral with two opposite angles equal 90, ZIMO - 09
Given a quadrilateral
A
B
C
D
ABCD
A
BC
D
with \angle B\equal{}\angle D\equal{}90^{\circ}. Point
M
M
M
is chosen on segment
A
B
AB
A
B
so taht AD\equal{}AM. Rays
D
M
DM
D
M
and
C
B
CB
CB
intersect at point
N
N
N
. Points
H
H
H
and
K
K
K
are feet of perpendiculars from points
D
D
D
and
C
C
C
to lines
A
C
AC
A
C
and
A
N
AN
A
N
, respectively. Prove that \angle MHN\equal{}\angle MCK.
1
2
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Pretty simple diophantine equation, ZIMO-09
Find all pairs of integers
(
x
,
y
)
(x,y)
(
x
,
y
)
, such that x^2 \minus{} 2009y \plus{} 2y^2 \equal{} 0
Four concyclic points on each of two parabola's, ZIMO - 09
On the plane, a Cartesian coordinate system is chosen. Given points
A
1
,
A
2
,
A
3
,
A
4
A_1,A_2,A_3,A_4
A
1
,
A
2
,
A
3
,
A
4
on the parabola y \equal{} x^2, and points
B
1
,
B
2
,
B
3
,
B
4
B_1,B_2,B_3,B_4
B
1
,
B
2
,
B
3
,
B
4
on the parabola y \equal{} 2009x^2. Points
A
1
,
A
2
,
A
3
,
A
4
A_1,A_2,A_3,A_4
A
1
,
A
2
,
A
3
,
A
4
are concyclic, and points
A
i
A_i
A
i
and
B
i
B_i
B
i
have equal abscissas for each i \equal{} 1,2,3,4. Prove that points
B
1
,
B
2
,
B
3
,
B
4
B_1,B_2,B_3,B_4
B
1
,
B
2
,
B
3
,
B
4
are also concyclic.