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National and Regional Contests
Russia Contests
Russian Team Selection Tests
Russian TST 2018
Russian TST 2018
Part of
Russian Team Selection Tests
Subcontests
(4)
P4
2
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Generalized Moldova TST
Let
a
1
,
…
,
a
n
+
1
a_1,\ldots,a_{n+1}
a
1
,
…
,
a
n
+
1
be positive real numbers satisfying
1
/
(
a
1
+
1
)
+
⋯
+
1
/
(
a
n
+
1
+
1
)
=
n
1/(a_1+1)+\cdots+1/(a_{n+1}+1)=n
1/
(
a
1
+
1
)
+
⋯
+
1/
(
a
n
+
1
+
1
)
=
n
. Prove that
∑
i
=
1
n
+
1
∏
j
≠
i
a
j
n
⩽
n
+
1
n
.
\sum_{i=1}^{n+1}\prod_{j\neq i}\sqrt[n]{a_j}\leqslant\frac{n+1}{n}.
i
=
1
∑
n
+
1
j
=
i
∏
n
a
j
⩽
n
n
+
1
.
Peter is arranging his items
The natural numbers
k
⩾
n
k \geqslant n
k
⩾
n
are given. Peter has
n
n{}
n
objects and
N
N{}
N
special ways in which he likes to lay them out in a row from left to right. He noticed that for any non-empty subset
A
A{}
A
of these objects containing
∣
A
∣
⩽
k
|A| \leqslant k
∣
A
∣
⩽
k
objects, and any element
a
∈
A
a\in A
a
∈
A
, there are exactly
N
/
∣
A
∣
N/|A|
N
/∣
A
∣
special ways for which element
a
a{}
a
is the leftmost in the set
A
A{}
A
. Prove that, under the same conditions on
A
A{}
A
and
a
a{}
a
, for any integer
m
=
1
,
2
,
…
,
∣
A
∣
m =1,2,\ldots,|A|
m
=
1
,
2
,
…
,
∣
A
∣
there are exactly
N
/
∣
A
∣
N/|A|
N
/∣
A
∣
special ways for which the element
a
a{}
a
is the
m
th
m^{\text{th}}
m
th
from the left in the set
A
A{}
A
.
P2
4
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P1
8
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P3
6
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