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4-variable inequality with square root

Source: 2023 Austrian Federal Competition For Advanced Students, Part 1 p1

May 4, 2023
algebrainequalities proposed

Problem Statement

Let a,b,c,da, b, c, d be positive reals strictly smaller than 11, such that a+b+c+d=2a+b+c+d=2. Prove that (1−a)(1−b)(1−c)(1−d)≤ac+bd2.\sqrt{(1-a)(1-b)(1-c)(1-d)} \leq \frac{ac+bd}{2}.
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