MathDB
Problems
Contests
International Contests
Tournament Of Towns
1982 Tournament Of Towns
(020) 1
(020) 1
Part of
1982 Tournament Of Towns
Problems
(1)
TOT 020 1982 Spring S1 sum {x_k}{x_{k-1}+x_{k+1}} >=2, sharp inequality
Source:
8/17/2019
(a) Prove that for any positive numbers
x
1
,
x
2
,
.
.
.
,
x
k
x_1,x_2,...,x_k
x
1
,
x
2
,
...
,
x
k
(
k
>
3
k > 3
k
>
3
),
x
1
x
k
+
x
2
+
x
2
x
1
+
x
3
+
.
.
.
+
x
k
x
k
−
1
+
x
1
≥
2
\frac{x_1}{x_k+x_2}+ \frac{x_2}{x_1+x_3}+...+\frac{x_k}{x_{k-1}+x_1}\ge 2
x
k
+
x
2
x
1
+
x
1
+
x
3
x
2
+
...
+
x
k
−
1
+
x
1
x
k
≥
2
(b) Prove that for every
k
k
k
this inequality cannot be sharpened, i.e. prove that for every given
k
k
k
it is not possible to change the number
2
2
2
in the right hand side to a greater number in such a way that the inequality remains true for every choice of positive numbers
x
1
,
x
2
,
.
.
.
,
x
k
x_1,x_2,...,x_k
x
1
,
x
2
,
...
,
x
k
.(A Prokopiev)
inequalities
algebra