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National and Regional Contests
Bulgaria Contests
Bulgaria JBMO Team Selection Test
2017 Bulgaria JBMO TST
3
4-variable inequality
4-variable inequality
Source: Bulgaria JBMO TST 2017 P3 day 2
August 26, 2019
Problem Statement
Prove that for all positive real
m
,
n
,
p
,
q
m, n, p, q
m
,
n
,
p
,
q
and
t
=
m
+
n
+
p
+
q
2
t=\frac{m+n+p+q}{2}
t
=
2
m
+
n
+
p
+
q
,
m
t
+
n
+
p
+
q
+
n
t
+
m
+
p
+
q
+
p
t
+
m
+
n
+
q
+
q
t
+
m
+
n
+
p
≥
4
5
.
\frac{m}{t+n+p+q} +\frac{n}{t+m+p+q} +\frac{p} {t+m+n+q}+\frac{q}{t+m+n+p} \geq \frac{4}{5}.
t
+
n
+
p
+
q
m
+
t
+
m
+
p
+
q
n
+
t
+
m
+
n
+
q
p
+
t
+
m
+
n
+
p
q
≥
5
4
.
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