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Ineq from Abel competition 2020-2021

Source: Abel Math Competition 2020-2021 Problem 2 part (b)

May 27, 2021
abelinequalitiesalgebra

Problem Statement

If a1,⋯ ,ana_1,\cdots,a_n and b1,⋯ ,bnb_1,\cdots,b_n are real numbers satisfying a12+⋯+an2≤1a_1^2+\cdots+a_n^2 \le 1 and b12+⋯+bn2≤1b_1^2+\cdots+b_n^2 \le 1 , show that: (1−(a12+⋯+an2))(1−(b12+⋯+bn2))≤(1−(a1b1+⋯+anbn))2(1-(a_1^2+\cdots+a_n^2))(1-(b_1^2+\cdots+b_n^2)) \le (1-(a_1b_1+\cdots+a_nb_n))^2
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