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Austrian-Polish
1998 Austrian-Polish Competition
1
(x_1y_1 + x_2y_2 - 1)^2 >= (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)
(x_1y_1 + x_2y_2 - 1)^2 >= (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)
Source: Austrian - Polish 1998 APMC
May 4, 2020
inequalities
algebra
Problem Statement
Let
x
1
,
x
2
,
y
1
,
y
2
x_1, x_2,y _1,y_2
x
1
,
x
2
,
y
1
,
y
2
be real numbers such that
x
1
2
+
x
2
2
≤
1
x_1^2 + x_2^2 \le 1
x
1
2
+
x
2
2
≤
1
. Prove the inequality
(
x
1
y
1
+
x
2
y
2
−
1
)
2
≥
(
x
1
2
+
x
2
2
−
1
)
(
y
1
2
+
y
2
2
−
1
)
(x_1y_1 + x_2y_2 - 1)^2 \ge (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)
(
x
1
y
1
+
x
2
y
2
−
1
)
2
≥
(
x
1
2
+
x
2
2
−
1
)
(
y
1
2
+
y
2
2
−
1
)
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