9th Eötvös Competition Problems 1902

1.  Let p(x) = ax2 + bx + c be a quadratic with real coefficients. Show that we can find reals d, e, f so that p(x) = d/2 x(x - 1) + ex + f, and that p(n) is always integral for integral n iff d, e, f are integers. 2.  P is a variable point outside the fixed sphere S with center O. Show that the surface area of the sphere center P radius PO which lies