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2024-IMOC
A8
A8
Part of
2024-IMOC
Problems
(1)
Old problem in IMOC
Source: 2024 IMOC A8 (Night 6)
8/8/2024
a
a
a
,
b
b
b
,
c
c
c
are three distinct real numbers, given
λ
>
0
\lambda >0
λ
>
0
. Proof that
1
+
λ
2
a
2
b
2
(
a
−
b
)
2
+
1
+
λ
2
b
2
c
2
(
b
−
c
)
2
+
1
+
λ
2
c
2
a
2
(
c
−
a
)
2
≥
3
2
λ
.
\frac{1+ \lambda ^2a^2b^2}{(a-b)^2}+\frac{1+ \lambda ^2b^2c^2}{(b-c)^2}+\frac{1+ \lambda ^2c^2a^2}{(c-a)^2} \geq \frac 32 \lambda.
(
a
−
b
)
2
1
+
λ
2
a
2
b
2
+
(
b
−
c
)
2
1
+
λ
2
b
2
c
2
+
(
c
−
a
)
2
1
+
λ
2
c
2
a
2
≥
2
3
λ
.
Old problem, can be found [url=https://artofproblemsolving.com/community/c6h588854p3487434]here. Double post to have a cleaner thread for collection (as the original one contains a messy quote)
algebra
inequalities