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Problems
Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad
2005 All-Russian Olympiad
2005 All-Russian Olympiad
Part of
All-Russian Olympiad
Subcontests
(4)
2
3
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Oleg always choose two cells
Lesha put numbers from 1 to
2
2
2
22^2
2
2
2
into cells of
22
×
22
22\times 22
22
×
22
board. Can Oleg always choose two cells, adjacent by the side or by vertex, the sum of numbers in which is divisible by 4?
x^2-S(A)x+S(B)=0 has integral roots
Find the number of subsets
A
⊂
M
=
{
2
0
,
2
1
,
2
2
,
…
,
2
2005
}
A\subset M=\{2^0,\,2^1,\,2^2,\dots,2^{2005}\}
A
⊂
M
=
{
2
0
,
2
1
,
2
2
,
…
,
2
2005
}
such that equation
x
2
−
S
(
A
)
x
+
S
(
B
)
=
0
x^2-S(A)x+S(B)=0
x
2
−
S
(
A
)
x
+
S
(
B
)
=
0
has integral roots, where
S
(
M
)
S(M)
S
(
M
)
is the sum of all elements of
M
M
M
, and
B
=
M
∖
A
B=M\setminus A
B
=
M
∖
A
(
A
A
A
and
B
B
B
are not empty).
12 rectangular parallelepipeds with edges parallel to axes
Do there exist 12 rectangular parallelepipeds
P
1
,
P
2
,
…
,
P
12
P_1,\,P_2,\ldots,P_{12}
P
1
,
P
2
,
…
,
P
12
with edges parallel to coordinate axes
O
X
,
O
Y
,
O
Z
OX,\,OY,\,OZ
OX
,
O
Y
,
OZ
such that
P
i
P_i
P
i
and
P
j
P_j
P
j
have a common point iff
i
≠
j
±
1
i\ne j\pm 1
i
=
j
±
1
modulo 12?
4
5
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3
3
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ARO 2005 inequality
Given three reals
a
1
,
a
2
,
a
3
>
1
,
S
=
a
1
+
a
2
+
a
3
a_1,\,a_2,\,a_3>1,\,S=a_1+a_2+a_3
a
1
,
a
2
,
a
3
>
1
,
S
=
a
1
+
a
2
+
a
3
. Provided
a
i
2
a
i
−
1
>
S
{a_i^2\over a_i-1}>S
a
i
−
1
a
i
2
>
S
for every
i
=
1
,
2
,
3
i=1,\,2,\,3
i
=
1
,
2
,
3
prove that
1
a
1
+
a
2
+
1
a
2
+
a
3
+
1
a
3
+
a
1
>
1.
\frac{1}{a_1+a_2}+\frac{1}{a_2+a_3}+\frac{1}{a_3+a_1}>1.
a
1
+
a
2
1
+
a
2
+
a
3
1
+
a
3
+
a
1
1
>
1.
the midpoint of PQ lies on A'B'
We have an acute-angled triangle
A
B
C
ABC
A
BC
, and
A
A
′
,
B
B
′
AA',BB'
A
A
′
,
B
B
′
are its altitudes. A point
D
D
D
is chosen on the arc
A
C
B
ACB
A
CB
of the circumcircle of
A
B
C
ABC
A
BC
. If
P
=
A
A
′
∩
B
D
,
Q
=
B
B
′
∩
A
D
P=AA'\cap BD,Q=BB'\cap AD
P
=
A
A
′
∩
B
D
,
Q
=
B
B
′
∩
A
D
, show that the midpoint of
P
Q
PQ
PQ
lies on
A
′
B
′
A'B'
A
′
B
′
.
minimal number of questions necessary to find all numbers
Given 2005 distinct numbers
a
1
,
a
2
,
…
,
a
2005
a_1,\,a_2,\dots,a_{2005}
a
1
,
a
2
,
…
,
a
2005
. By one question, we may take three different indices
1
≤
i
<
j
<
k
≤
2005
1\le i<j<k\le 2005
1
≤
i
<
j
<
k
≤
2005
and find out the set of numbers
{
a
i
,
a
j
,
a
k
}
\{a_i,\,a_j,\,a_k\}
{
a
i
,
a
j
,
a
k
}
(unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers
a
i
a_i
a
i
.
1
5
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