MathDB
Problems
Contests
International Contests
IMO Shortlist
2015 IMO Shortlist
A1
A1
Part of
2015 IMO Shortlist
Problems
(1)
n-variable inequality
Source: 2015 IMO Shortlist A1, Original 2015 IMO #5
7/7/2016
Suppose that a sequence
a
1
,
a
2
,
…
a_1,a_2,\ldots
a
1
,
a
2
,
…
of positive real numbers satisfies
a
k
+
1
≥
k
a
k
a
k
2
+
(
k
−
1
)
a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}
a
k
+
1
≥
a
k
2
+
(
k
−
1
)
k
a
k
for every positive integer
k
k
k
. Prove that
a
1
+
a
2
+
…
+
a
n
≥
n
a_1+a_2+\ldots+a_n\geq n
a
1
+
a
2
+
…
+
a
n
≥
n
for every
n
≥
2
n\geq2
n
≥
2
.
algebra
IMO Shortlist
Sequence
Inequality
induction