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sum a/(a^2+7) <= 3/8 if ab + bc + ca = 3 for a,b,c>=0
sum a/(a^2+7) <= 3/8 if ab + bc + ca = 3 for a,b,c>=0
Source: 2018 Romanian NMO grade VIII P4
September 4, 2024
algebra
inequalities
Problem Statement
Let
a
,
b
,
c
≥
0
a, b, c \ge 0
a
,
b
,
c
≥
0
so that
a
b
+
b
c
+
c
a
=
3
ab + bc + ca = 3
ab
+
b
c
+
c
a
=
3
. Prove that:
a
a
2
+
7
+
b
b
2
+
7
+
c
c
2
+
7
≤
3
8
\frac{a}{a^2+7}+\frac{b}{b^2+7}+\frac{c}{c^2+7}\le \frac38
a
2
+
7
a
+
b
2
+
7
b
+
c
2
+
7
c
≤
8
3
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