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Baltic Way
2012 Baltic Way
2
Inequality with Max
Inequality with Max
Source: 2012 Baltic Way, Problem 2
November 22, 2012
inequalities
inequalities unsolved
Problem Statement
Let
a
a
a
,
b
b
b
,
c
c
c
be real numbers. Prove that
a
b
+
b
c
+
c
a
+
max
{
∣
a
−
b
∣
,
∣
b
−
c
∣
,
∣
c
−
a
∣
}
≤
1
+
1
3
(
a
+
b
+
c
)
2
.
ab + bc + ca + \max\{|a - b|, |b - c|, |c - a|\} \le 1 + \frac{1}{3} (a + b + c)^2.
ab
+
b
c
+
c
a
+
max
{
∣
a
−
b
∣
,
∣
b
−
c
∣
,
∣
c
−
a
∣
}
≤
1
+
3
1
(
a
+
b
+
c
)
2
.
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