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2017 Auckland Mathematical Olympiad
2
(1 - x_1)(1 - x_2)(1 - x_3) >= 1/2 if x_1+x_2+x_3<=1/2 (2017 Auckland MO S2)
(1 - x_1)(1 - x_2)(1 - x_3) >= 1/2 if x_1+x_2+x_3<=1/2 (2017 Auckland MO S2)
Source:
November 15, 2021
algebra
inequalities
Problem Statement
The sum of the three nonnegative real numbers
x
1
,
x
2
,
x
3
x_1, x_2, x_3
x
1
,
x
2
,
x
3
is not greater than
1
2
\frac12
2
1
. Prove that
(
1
−
x
1
)
(
1
−
x
2
)
(
1
−
x
3
)
≥
1
2
(1 - x_1)(1 - x_2)(1 - x_3) \ge \frac12
(
1
−
x
1
)
(
1
−
x
2
)
(
1
−
x
3
)
≥
2
1
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