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National and Regional Contests
China Contests
China Team Selection Test
2004 China Team Selection Test
3
China TST 2004 product inequality
China TST 2004 product inequality
Source: China Team Selection Test 2004, Day 1, Problem 3
October 14, 2005
inequalities
LaTeX
inequalities unsolved
Problem Statement
Let
k
≥
2
,
1
<
n
1
<
n
2
<
…
<
n
k
k \geq 2, 1 < n_1 < n_2 < \ldots < n_k
k
≥
2
,
1
<
n
1
<
n
2
<
…
<
n
k
are positive integers,
a
,
b
∈
Z
+
a,b \in \mathbb{Z}^+
a
,
b
∈
Z
+
satisfy
∏
i
=
1
k
(
1
−
1
n
i
)
≤
a
b
<
∏
i
=
1
k
−
1
(
1
−
1
n
i
)
\prod^k_{i=1} \left( 1 - \frac{1}{n_i} \right) \leq \frac{a}{b} < \prod^{k-1}_{i=1} \left( 1 - \frac{1}{n_i} \right)
i
=
1
∏
k
(
1
−
n
i
1
)
≤
b
a
<
i
=
1
∏
k
−
1
(
1
−
n
i
1
)
Prove that:
∏
i
=
1
k
n
i
≥
(
4
⋅
a
)
2
k
−
1
.
\prod^k_{i=1} n_i \geq (4 \cdot a)^{2^k - 1}.
i
=
1
∏
k
n
i
≥
(
4
⋅
a
)
2
k
−
1
.
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