MathDB
Problems
Contests
International Contests
IMO Shortlist
2021 IMO Shortlist
A7
Weird exponent, but ok
Weird exponent, but ok
Source: 2021 ISL A7
July 12, 2022
ISL 2021
algebra
Inequality
inequalities
Problem Statement
Let
n
⩾
1
n\geqslant 1
n
⩾
1
be an integer, and let
x
0
,
x
1
,
…
,
x
n
+
1
x_0,x_1,\ldots,x_{n+1}
x
0
,
x
1
,
…
,
x
n
+
1
be
n
+
2
n+2
n
+
2
non-negative real numbers that satisfy
x
i
x
i
+
1
−
x
i
−
1
2
⩾
1
x_ix_{i+1}-x_{i-1}^2\geqslant 1
x
i
x
i
+
1
−
x
i
−
1
2
⩾
1
for all
i
=
1
,
2
,
…
,
n
.
i=1,2,\ldots,n.
i
=
1
,
2
,
…
,
n
.
Show that
x
0
+
x
1
+
⋯
+
x
n
+
x
n
+
1
>
(
2
n
3
)
3
/
2
.
x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.
x
0
+
x
1
+
⋯
+
x
n
+
x
n
+
1
>
(
3
2
n
)
3/2
.
Pakawut Jiradilok and Wijit Yangjit, Thailand
Back to Problems
View on AoPS