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2023 JBMO Shortlist
A1
JBMO Shortlist 2023 A1
JBMO Shortlist 2023 A1
Source: JBMO Shortlist 2023, A1
June 28, 2024
inequalities
JBMO
JBMO Shortlist
algebra
Problem Statement
Prove that for all positive real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
,
2
(
a
+
b
)
(
c
+
d
)
+
(
b
+
c
)
(
a
+
d
)
≤
1
(
a
+
c
)
(
b
+
d
)
+
4
a
c
+
1
(
a
+
c
)
(
b
+
d
)
+
4
b
d
\frac{2}{(a+b)(c+d)+(b+c)(a+d)} \leq \frac{1}{(a+c)(b+d)+4ac}+\frac{1}{(a+c)(b+d)+4bd}
(
a
+
b
)
(
c
+
d
)
+
(
b
+
c
)
(
a
+
d
)
2
≤
(
a
+
c
)
(
b
+
d
)
+
4
a
c
1
+
(
a
+
c
)
(
b
+
d
)
+
4
b
d
1
and determine when equality occurs.
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