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$y=x^3-px$ so as ussual we find the slope of any tangent to the curve at a point $x_0$


$3x_0^2-p$ so the equation of the tangent is


if we draw a tangent from $(x_1,y_1)$ where $y_1\ne x_1^3-px_1 $ , our interest is how many such different pairs can we find that are tangent to the curve... obxiously $x_0$ is unique in the sense that only one such point exist but our interest is what about other points on the will be appreciated

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In this generality even by a CAS the problem is difficult ... – Tony Piccolo Nov 16 '13 at 17:10
but suprisingly this was asked in a high school contest problem IMO ,[imo problem,39,1972][1] [1]:… – Jonas12 Nov 16 '13 at 20:23

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