8th Putnam 1948 Problem B1 p(x) is a cubic polynomial with roots α, β, γ and p'(x) divides p(2x). Find the ratios α : β : γ.
p(x) = (x - α)(x - β)(x - γ). Hence p'(x) = 3x 2 - 2(α+β+γ)x + βγ+γα+aβ. This must have roots α/2, β/2. Hence 2(α+β+γ)/3 = (α+β)/2 and αβ/4 = (βγ+γα+αβ)/3. This leads to α = (-2 + 2i/√3)γ, β = (-2 - 2i/√3)γ. 8th Putnam 1948 © John Scholes jscholes@kalva.demon.co.uk 11 Mar 2002