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

$\dfrac{dy}{dx}=3x^2-p$

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

$y=-x_0^3+px_0+(3x_0^2-p)(x-x_0)\\y=x(3x_0^2-p)-2x_0^3$

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 graph....help 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]: books.google.nl/… –  Jonas12 Nov 16 '13 at 20:23
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