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IMC
2010 IMC
2
IMC 2010, Problem 2, Day 2
IMC 2010, Problem 2, Day 2
Source:
July 27, 2010
inequalities
induction
inequalities unsolved
Problem Statement
Let
a
0
,
a
1
,
…
,
a
n
a_0,a_1,\dots,a_n
a
0
,
a
1
,
…
,
a
n
be positive real numbers such that
a
k
+
1
−
a
k
≥
1
a_{k+1}-a_k \geq 1
a
k
+
1
−
a
k
≥
1
for all
k
=
0
,
1
,
…
,
n
−
1.
k=0,1,\dots,n-1.
k
=
0
,
1
,
…
,
n
−
1.
Prove that
1
+
1
a
0
(
1
+
1
a
1
−
a
0
)
⋯
(
1
+
1
a
n
−
a
0
)
≤
(
1
+
1
a
0
)
(
1
+
1
a
1
)
⋯
(
1
+
1
a
n
)
.
1+\frac{1}{a_0} \left( 1+\frac1{a_1-a_0}\right)\cdots\left(1+\frac1{a_n-a_0}\right)\leq \left(1+\frac1{a_0}\right) \left(1+\frac1{a_1}\right)\cdots \left(1+\frac1{a_n}\right).
1
+
a
0
1
(
1
+
a
1
−
a
0
1
)
⋯
(
1
+
a
n
−
a
0
1
)
≤
(
1
+
a
0
1
)
(
1
+
a
1
1
)
⋯
(
1
+
a
n
1
)
.
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