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Austrian-Polish
1996 Austrian-Polish Competition
4
- 1<=xy + yz + zt + tx <=0 if x + y + z +t = 0 and x^2+ y^2+ z^2+t^2 = 1
- 1<=xy + yz + zt + tx <=0 if x + y + z +t = 0 and x^2+ y^2+ z^2+t^2 = 1
Source: Austrian - Polish 1996 APMC
May 3, 2020
inequalities
algebra
Problem Statement
Real numbers
x
,
y
,
z
,
t
x,y,z, t
x
,
y
,
z
,
t
satisfy
x
+
y
+
z
+
t
=
0
x + y + z +t = 0
x
+
y
+
z
+
t
=
0
and
x
2
+
y
2
+
z
2
+
t
2
=
1
x^2+ y^2+ z^2+t^2 = 1
x
2
+
y
2
+
z
2
+
t
2
=
1
. Prove that
−
1
≤
x
y
+
y
z
+
z
t
+
t
x
≤
0
- 1 \le xy + yz + zt + tx \le 0
−
1
≤
x
y
+
yz
+
z
t
+
t
x
≤
0
.
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