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Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
1999 German National Olympiad
5
5
Part of
1999 German National Olympiad
Problems
(1)
|x-y|+|y-z|+|z-x| \le a \sqrt{x^2 +y^2 +z^2}
Source: Germany 1999 p5
2/23/2020
Consider the following inequality for real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
:
∣
x
−
y
∣
+
∣
y
−
z
∣
+
∣
z
−
x
∣
≤
a
x
2
+
y
2
+
z
2
|x-y|+|y-z|+|z-x| \le a \sqrt{x^2 +y^2 +z^2}
∣
x
−
y
∣
+
∣
y
−
z
∣
+
∣
z
−
x
∣
≤
a
x
2
+
y
2
+
z
2
. (a) Prove that the inequality is valid for
a
=
2
2
a = 2\sqrt2
a
=
2
2
(b) Assuming that
x
,
y
,
z
x,y,z
x
,
y
,
z
are nonnegative, show that the inequality is also valid for
a
=
2
a = 2
a
=
2
.
inequalities
algebra