2

Part of 1950 Miklós Schweitzer

Problems(2)

Miklos Schweitzer 1950_2

Source: first round of 1950

Consider three different planes and consider also one point on each of them. Give necessary and sufficient conditions for the existence of a quadratic which passes through the given points and whose tangent-plane at each of these points is the respective given plane.

quadraticsgeometry proposedgeometry

Miklos Schweitzer 1950_2

Source: second part of 1950

Show that there exists a positive constant

c

with the following property: To every positive irrational

\alpha

, there can be found infinitely many fractions

\frac{p}{q}

with (p,q)\equal{}1 satisfying \left|\alpha\minus{}\frac{p}{q}\right|\le \frac{c}{q^2}

continued fractionnumber theory proposednumber theory