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2019 Silk Road
2
(a_1 + 1)(a_2 + 2)... (a_ {99} +99)>=100! in positive real under conditions
(a_1 + 1)(a_2 + 2)... (a_ {99} +99)>=100! in positive real under conditions
Source: SRMC 2019 P2
July 16, 2019
algebra
inequalities
Problem Statement
Let
a
1
,
a_1,
a
1
,
a
2
,
a_2,
a
2
,
…
,
\ldots,
…
,
a
99
a_ {99}
a
99
be positive real numbers such that
i
a
j
+
j
a
i
≥
i
+
j
ia_j + ja_i \ge i + j
i
a
j
+
j
a
i
≥
i
+
j
for all
1
≤
i
<
j
≤
99.
1 \le i <j \le 99.
1
≤
i
<
j
≤
99.
Prove , that
(
a
1
+
1
)
(
a
2
+
2
)
…
(
a
99
+
99
)
≥
100
!
(a_1 + 1) (a_2 + 2) \ldots (a_ {99} +99) \ge 100!
(
a
1
+
1
)
(
a
2
+
2
)
…
(
a
99
+
99
)
≥
100
!
.
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