If you take a conic through 5 rational points on a quartic curve, then will at least one of the remaining 3 points also be rational ? 



rational points on degree 4 curve
No. Let the conic be the union of lines L\_1 and L\_2, one of which intersects the quartic Q in four rational points. Now the intersection of L\_2 with Q is parametrized by the roots of a quartic equation which can be whatever it wants; in particular it can have one root rational and the rest not.


