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2023 JBMO Shortlist
A4
JBMO Shortlist 2023 A4
JBMO Shortlist 2023 A4
Source: JBMO Shortlist 2023, A4
June 28, 2024
inequalities
JBMO
JBMO Shortlist
algebra
Problem Statement
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive real numbers with
a
b
c
d
=
1
abcd=1
ab
c
d
=
1
. Prove that
a
b
+
c
+
d
2
+
a
3
+
b
c
+
d
+
a
2
+
b
3
+
c
d
+
a
+
b
2
+
c
3
+
d
a
+
b
+
c
2
+
d
3
≤
2
\sqrt{\frac{a}{b+c+d^2+a^3}}+\sqrt{\frac{b}{c+d+a^2+b^3}}+\sqrt{\frac{c}{d+a+b^2+c^3}}+\sqrt{\frac{d}{a+b+c^2+d^3}} \leq 2
b
+
c
+
d
2
+
a
3
a
+
c
+
d
+
a
2
+
b
3
b
+
d
+
a
+
b
2
+
c
3
c
+
a
+
b
+
c
2
+
d
3
d
≤
2
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