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Russian Team Selection Tests
Russian TST 2018
P4
Generalized Moldova TST
Generalized Moldova TST
Source: Russian TST 2018, Day 8 P4 (Groups A & B)
March 30, 2023
algebra
inequalities
Problem Statement
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
.
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