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National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2023 Moldova Team Selection Test
9
9
Part of
2023 Moldova Team Selection Test
Problems
(1)
E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2}
Source: Moldova TST 2023
4/9/2023
Let
n
n
n
(
n
≥
2
)
(n\geq2)
(
n
≥
2
)
be an integer. Find the greatest possible value of the expression
E
=
a
1
1
+
a
1
2
+
a
2
1
+
a
2
2
+
…
+
a
n
1
+
a
n
2
E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2}
E
=
1
+
a
1
2
a
1
+
1
+
a
2
2
a
2
+
…
+
1
+
a
n
2
a
n
if the positive real numbers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
satisfy
a
1
+
a
2
+
…
+
a
n
=
n
2
.
a_1+a_2+\ldots+a_n=\frac{n}{2}.
a
1
+
a
2
+
…
+
a
n
=
2
n
.
What are the values of
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
when the greatest value is achieved?
inequalities