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Casey variable modulus expression

Source: Romanian District Olympiad, Grade X, Problem 3

October 5, 2018
complex numbersalgebra

Problem Statement

Let α,β \alpha ,\beta be real numbers. Find the greatest value of the expression ∣αx+βy∣+∣αx−βy∣ |\alpha x +\beta y| +|\alpha x-\beta y| in each of the following cases:
a) x,y∈R x,y\in \mathbb{R} and ∣x∣,∣y∣≤1 |x|,|y|\le 1 b) x,y∈C x,y\in \mathbb{C} and ∣x∣,∣y∣≤1 |x|,|y|\le 1
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