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2022 Turkey MO (2nd round)
3
3
Part of
2022 Turkey MO (2nd round)
Problems
(1)
max # of pairs satisying a^2+b>=1/2021
Source: Turkey National Mathematical Olympiad 2022 P3
12/23/2022
Let
a
1
,
a
2
,
⋯
,
a
2022
a_1, a_2, \cdots, a_{2022}
a
1
,
a
2
,
⋯
,
a
2022
be nonnegative real numbers such that
a
1
+
a
2
+
⋯
+
a
2022
=
1
a_1+a_2+\cdots +a_{2022}=1
a
1
+
a
2
+
⋯
+
a
2022
=
1
. Find the maximum number of ordered pairs
(
i
,
j
)
(i, j)
(
i
,
j
)
,
1
≤
i
,
j
≤
2022
1\leq i,j\leq 2022
1
≤
i
,
j
≤
2022
, satisfying
a
i
2
+
a
j
≥
1
2021
.
a_i^2+a_j\ge \frac 1{2021}.
a
i
2
+
a
j
≥
2021
1
.
inequalities