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Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2024 Moldova Team Selection Test
7
7
Part of
2024 Moldova Team Selection Test
Problems
(1)
Tricky Shapiro-like inequality
Source: Moldova TST 2024 P7
6/9/2024
Prove that
a
=
2
a=2
a
=
2
is the greatest real number for which the inequality:
x
1
x
n
+
x
2
+
x
2
x
1
+
x
3
+
⋯
+
x
n
x
n
−
1
+
x
1
≥
a
\frac{x_1}{x_n+x_2}+\frac{x_2}{x_1+x_3}+\dots+\frac{x_n}{x_{n-1}+x_1} \ge a
x
n
+
x
2
x
1
+
x
1
+
x
3
x
2
+
⋯
+
x
n
−
1
+
x
1
x
n
≥
a
holds true for any
n
≥
4
n \ge 4
n
≥
4
and any positive real numbers
x
1
,
x
2
,
…
,
x
n
x_1, x_2,\dots,x_n
x
1
,
x
2
,
…
,
x
n
.
inequalities
TST