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Tuymaada Olympiad
2021 Tuymaada Olympiad
3
Inequality back to business again!
Inequality back to business again!
Source: Tuymaada 2021 Senior P3
July 28, 2021
inequalities
algebra
Problem Statement
Positive real numbers
a
1
,
…
,
a
k
,
b
1
,
…
,
b
k
a_1, \dots, a_k, b_1, \dots, b_k
a
1
,
…
,
a
k
,
b
1
,
…
,
b
k
are given. Let
A
=
∑
i
=
1
k
a
i
,
B
=
∑
i
=
1
k
b
i
A = \sum_{i = 1}^k a_i, B = \sum_{i = 1}^k b_i
A
=
∑
i
=
1
k
a
i
,
B
=
∑
i
=
1
k
b
i
. Prove the inequality
(
∑
i
=
1
k
a
i
b
i
a
i
B
+
b
i
A
−
1
)
2
≥
∑
i
=
1
k
a
i
2
a
i
B
+
b
i
A
⋅
∑
i
=
1
k
b
i
2
a
i
B
+
b
i
A
.
\left( \sum_{i = 1}^k \frac{a_i b_i}{a_i B + b_i A} - 1 \right)^2 \ge \sum_{i = 1}^k \frac{a_i^2}{a_i B + b_i A} \cdot \sum_{i = 1}^k \frac{b_i^2}{a_i B + b_i A}.
(
i
=
1
∑
k
a
i
B
+
b
i
A
a
i
b
i
−
1
)
2
≥
i
=
1
∑
k
a
i
B
+
b
i
A
a
i
2
⋅
i
=
1
∑
k
a
i
B
+
b
i
A
b
i
2
.
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