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Viete and polynomials

Source: Nigeria Olympiad senior mathematics round 2 2020 no 1

January 18, 2020
Algebra Polynomialsalgebrapolynomial

Problem Statement

Let kk be a real number. Define on the set of reals the operation x∗yx*y= xyx+y+k\frac{xy}{x+y+k} whenever x+yx+y does not equal −k-k. Let x1<x2<x3<x4x_1<x_2<x_3<x_4 be the roots of t4=27(t2+t+1)t^4=27(t^2+t+1).suppose that [(x1∗x2)∗x3]∗x4=1[(x_1*x_2)*x_3]*x_4=1. Find all possible values of kk
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