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2001 USAMO
3
Inequality with a^2+b^2+c^2+abc=4
Inequality with a^2+b^2+c^2+abc=4
Source: USAMO 2001 #3
June 27, 2004
inequalities
symmetry
trigonometry
AMC
USA(J)MO
USAMO
quadratics
Problem Statement
Let
a
,
b
,
c
≥
0
a, b, c \geq 0
a
,
b
,
c
≥
0
and satisfy
a
2
+
b
2
+
c
2
+
a
b
c
=
4.
a^2+b^2+c^2 +abc = 4 .
a
2
+
b
2
+
c
2
+
ab
c
=
4.
Show that
0
≤
a
b
+
b
c
+
c
a
−
a
b
c
≤
2.
0 \le ab + bc + ca - abc \leq 2.
0
≤
ab
+
b
c
+
c
a
−
ab
c
≤
2.
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