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Balkan MO Shortlist
2022 Balkan MO Shortlist
A2
Inequality with n terms
Inequality with n terms
Source: BMO Shortlist 2022, A2
May 13, 2023
algebra
inequalities
Problem Statement
Let
k
>
1
k > 1{}
k
>
1
be a real number,
n
⩾
3
n\geqslant 3
n
⩾
3
be an integer, and
x
1
⩾
x
2
⩾
⋯
⩾
x
n
x_1 \geqslant x_2\geqslant\cdots\geqslant x_n
x
1
⩾
x
2
⩾
⋯
⩾
x
n
be positive real numbers. Prove that
x
1
+
k
x
2
x
2
+
x
3
+
x
2
+
k
x
3
x
3
+
x
4
+
⋯
+
x
n
+
k
x
1
x
1
+
x
2
⩾
n
(
k
+
1
)
2
.
\frac{x_1+kx_2}{x_2+x_3}+\frac{x_2+kx_3}{x_3+x_4}+\cdots+\frac{x_n+kx_1}{x_1+x_2}\geqslant\frac{n(k+1)}{2}.
x
2
+
x
3
x
1
+
k
x
2
+
x
3
+
x
4
x
2
+
k
x
3
+
⋯
+
x
1
+
x
2
x
n
+
k
x
1
⩾
2
n
(
k
+
1
)
.
Ilija Jovcheski
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