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Moldova Team Selection Test
2022 Moldova Team Selection Test
10
P\left(\frac{x_1}{x_2} \right)^2+P\left(\frac{x_2}{x_3} \right)^2+ ... +P\left(\
P\left(\frac{x_1}{x_2} \right)^2+P\left(\frac{x_2}{x_3} \right)^2+ ... +P\left(\
Source: Moldova TST 2022
April 1, 2022
inequalities
Problem Statement
Let
P
(
X
)
P(X)
P
(
X
)
be a polynomial with positive coefficients. Show that for every integer
n
≥
2
n \geq 2
n
≥
2
and every
n
n
n
positive numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2,..., x_n
x
1
,
x
2
,
...
,
x
n
the following inequality is true:
P
(
x
1
x
2
)
2
+
P
(
x
2
x
3
)
2
+
.
.
.
+
P
(
x
n
x
1
)
2
≥
n
⋅
P
(
1
)
2
.
P\left(\frac{x_1}{x_2} \right)^2+P\left(\frac{x_2}{x_3} \right)^2+ ... +P\left(\frac{x_n}{x_1} \right)^2 \geq n \cdot P(1)^2.
P
(
x
2
x
1
)
2
+
P
(
x
3
x
2
)
2
+
...
+
P
(
x
1
x
n
)
2
≥
n
⋅
P
(
1
)
2
.
When does the equality take place?
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