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Poland Contests
Polish Junior Math Olympiad
2024 Polish Junior Math Olympiad
2024 Polish Junior MO Finals
3
3
Part of
2024 Polish Junior MO Finals
Problems
(1)
A funny inequality with (possibly negative) real numbers
Source: 2024 Polish Junior Math Olympiad Finals P3
5/5/2024
Real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy
a
+
b
≠
0
a+b \ne 0
a
+
b
=
0
,
b
+
c
≠
0
b+c \ne 0
b
+
c
=
0
and
c
+
a
≠
0
c+a \ne 0
c
+
a
=
0
. Show that
(
a
2
c
a
+
b
+
b
2
a
b
+
c
+
c
2
b
c
+
a
)
⋅
(
b
2
c
a
+
b
+
c
2
a
b
+
c
+
a
2
b
c
+
a
)
≥
0.
\left(\frac{a^2c}{a+b}+\frac{b^2a}{b+c}+\frac{c^2b}{c+a}\right) \cdot \left(\frac{b^2c}{a+b}+\frac{c^2a}{b+c}+\frac{a^2b}{c+a}\right) \ge 0.
(
a
+
b
a
2
c
+
b
+
c
b
2
a
+
c
+
a
c
2
b
)
⋅
(
a
+
b
b
2
c
+
b
+
c
c
2
a
+
c
+
a
a
2
b
)
≥
0.
inequalities
inequalities proposed
algebra
algebra proposed