MathDB

1

Part of 2014 USAMO

Problems(1)

Inequality on a Quartic

Source: USAMO 2014, Problem 1

4/29/2014
Let aa, bb, cc, dd be real numbers such that b−d≥5b-d \ge 5 and all zeros x1,x2,x3,x_1, x_2, x_3, and x4x_4 of the polynomial P(x)=x4+ax3+bx2+cx+dP(x)=x^4+ax^3+bx^2+cx+d are real. Find the smallest value the product (x12+1)(x22+1)(x32+1)(x42+1)(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1) can take.
inequalitiesalgebrapolynomialfunctiondomainUSAMO