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min, max of (1-x_1)(1-y_1)+(1-x_2)(1-y_2) when x_1^2+x_2^2=y_1^2+y_2^2=2013

Source: 2013 Saudi Arabia IMO TST II p1

July 23, 2020
minmaxalgebrainequalities

Problem Statement

Find the maximum and the minimum values of S=(1−x1)(1−y1)+(1−x2)(1−y2)S = (1 - x_1)(1 -y_1) + (1 - x_2)(1 - y_2) for real numbers x1,x2,y1,y2x_1, x_2, y_1,y_2 with x12+x22=y12+y22=2013x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013.
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