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Eotvos Mathematical Competition (Hungary)
1939 Eotvos Mathematical Competition
1
(a_1 + a_2)(c_1 + c_2) >= (b_1 + b_2)^2
(a_1 + a_2)(c_1 + c_2) >= (b_1 + b_2)^2
Source: Eotvos 1939 p1
September 10, 2024
algebra
inequalities
Problem Statement
Let
a
1
a_1
a
1
,
a
2
a_2
a
2
,
b
1
b_1
b
1
,
b
2
b_2
b
2
,
c
1
c_1
c
1
and
c
2
c_2
c
2
be real numbers for which
a
1
a
2
>
0
a_1a_2 > 0
a
1
a
2
>
0
,
a
1
c
1
≥
b
1
2
a_1c_1 \ge b^2_1
a
1
c
1
≥
b
1
2
and
a
2
c
2
>
b
2
2
a_2c_2 > b^2_2
a
2
c
2
>
b
2
2
. Prove that
(
a
1
+
a
2
)
(
c
1
+
c
2
)
≥
(
b
1
+
b
2
)
2
(a_1 + a_2)(c_1 + c_2) \ge (b_1 + b_2)^2
(
a
1
+
a
2
)
(
c
1
+
c
2
)
≥
(
b
1
+
b
2
)
2
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