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Swedish Mathematical Competition
2017 Swedish Mathematical Competition
6
6
Part of
2017 Swedish Mathematical Competition
Problems
(1)
ax^2+by^2+cz^2>=0 if x+y+z=0 , a+b+c>= 0, ab+bc+ca >=0
Source: 2017 Swedish Mathematical Competition p6
5/1/2021
Let
a
,
b
,
c
,
x
,
y
,
z
a,b,c,x,y,z
a
,
b
,
c
,
x
,
y
,
z
be real numbers such that
x
+
y
+
z
=
0
x+y+z=0
x
+
y
+
z
=
0
,
a
+
b
+
c
≥
0
a+b+c\geq 0
a
+
b
+
c
≥
0
,
a
b
+
b
c
+
c
a
≥
0
ab+bc+ca \ge 0
ab
+
b
c
+
c
a
≥
0
. Prove that
a
x
2
+
b
y
2
+
c
z
2
≥
0
ax^2+by^2+cz^2\ge 0
a
x
2
+
b
y
2
+
c
z
2
≥
0
algebra
inequalities