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2022 Switzerland Team Selection Test
5
Inequalities and lambda
Inequalities and lambda
Source: 2022 Switzerland IMO TST, Problem 5
August 7, 2022
inequalities
algebra
Problem Statement
Let
a
,
b
,
c
,
λ
a, b, c, \lambda
a
,
b
,
c
,
λ
be positive real numbers with
λ
≥
1
/
4
\lambda \geq 1/4
λ
≥
1/4
. Show that
a
b
2
+
λ
b
c
+
c
2
+
b
c
2
+
λ
c
a
+
a
2
+
c
a
2
+
λ
a
b
+
b
2
≥
3
λ
+
2
.
\frac{a}{\sqrt{b^2+\lambda bc+c^2}}+\frac{b}{\sqrt{c^2+\lambda ca+a^2}}+\frac{c}{\sqrt{a^2+\lambda ab+b^2}} \geq \frac{3}{\sqrt{\lambda +2}}.
b
2
+
λb
c
+
c
2
a
+
c
2
+
λ
c
a
+
a
2
b
+
a
2
+
λab
+
b
2
c
≥
λ
+
2
3
.
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