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4

Part of 1976 Putnam

Problems(2)

Putnam 1976 A4

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4/18/2022
Let rr be a root of P(x)=x3+ax2+bx−1=0P(x)=x^3+ax^2+bx-1=0 and r+1r+1 be a root of y3+cy2+dy+1=0,y^3+cy^2+dy+1=0, where a,b,ca,b,c and dd are integers. Also let P(x)P(x) be irreducible over the rational numbers. Express another root ss of P(x)=0P(x)=0 as a function of rr which does not explicitly involve a,b,ca,b,c or d.d.
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Putnam 1976 B4

Source:

4/20/2022
For a point PP on an ellipse, let dd be the distance from the center of the ellipse to the line tangent to the ellipse at P.P. Prove that (PF1)(PF2)d2(PF_1)(PF_2)d^2 is constant as PP varies on the ellipse, where PF1PF_1 and PF2PF_2 are distances from PP to the foci F1F_1 and F2F_2 of the ellipse.
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