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Problems
Contests
National and Regional Contests
Armenia Contests
Armenia National Math Olympiad
2011 Armenian Republican Olympiads
2011 Armenian Republican Olympiads
Part of
Armenia National Math Olympiad
Subcontests
(6)
Problem 6
1
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Problem 3, Day 2
Find the smallest
n
n
n
such that in an
8
×
8
8\times 8
8
×
8
chessboard any
n
n
n
cells contain two cells which are at least
3
3
3
knight moves apart from each other.
Problem 5
1
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Problem 2 Day 2
A quadrilateral
A
B
C
D
ABCD
A
BC
D
is such that
∠
A
=
∠
C
=
6
0
o
\angle A= \angle C=60^o
∠
A
=
∠
C
=
6
0
o
and
∠
B
=
10
0
o
\angle B=100^o
∠
B
=
10
0
o
. Let
O
1
O_1
O
1
and
O
2
O_2
O
2
be the centers of the incircles of triangles
A
B
D
ABD
A
B
D
and
C
B
D
CBD
CB
D
respectively. Find the angle between the lines
A
O
2
AO_2
A
O
2
and
C
O
1
CO_1
C
O
1
.
Problem 4
1
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Problem 1 Day 2
What is the maximal number of elements we can choose form the set
{
1
,
2
,
…
,
31
}
\{1, 2, \ldots, 31\}
{
1
,
2
,
…
,
31
}
, such that the sum of any two of them is not a perfect square.
Problem 3
1
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Problem 3 Day1
Find all integers
a
,
m
,
n
,
k
,
a, m, n, k,
a
,
m
,
n
,
k
,
such that
(
a
m
+
1
)
(
a
n
−
1
)
=
1
5
k
.
(a^m+1)(a^n-1)=15^k.
(
a
m
+
1
)
(
a
n
−
1
)
=
1
5
k
.
Problem 2
1
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Problem 2 Day1
Let a hexagone with a diameter
D
D
D
be given and let
d
>
D
2
.
d>\frac D 2.
d
>
2
D
.
On each side of the hexagon one constructs a isosceles triangle with two equal sides of length
d
d
d
. Prove that the sum of the areas of those isoscele triangles is greater than the area of a rhombus with side lengths
d
d
d
and a diagonal of length
D
D
D
. (The diameter of a polygon is the maximum of the lengths of all its sides and diagonals.)
Problem 1
1
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Problem 1
Does there exist a function
f
:
R
→
R
f\colon \mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that for any
x
>
y
,
x>y,
x
>
y
,
it satisfies
f
(
x
)
−
f
(
y
)
>
x
−
y
.
f(x)-f(y)>\sqrt{x-y}.
f
(
x
)
−
f
(
y
)
>
x
−
y
.