### 15th Putnam 1955

**Problem A5**

Given a parabola, construct the focus (with ruler and compass).

**Solution**

*Hard. Well, unless you know a lot about the geometry of the parabola.*

You need to know two facts:

(1) Rays from infinity come to a focus at the focus. In other words, if a line L parallel to the axis of the parabola meets it at X and the focus is F, then L and XF are equally inclined to the normal at X;

(2) The line joining the midpoints of two parallel chords is parallel to the axis.

The construction is then fairly obvious. Take two parallel chords. Join their midpoints and extend to meet the parabola at X. Take a line M through X parallel to the chords. Then M is a tangent. The line perpendicular to M through X is the normal. Hence find the line XF (although not yet F). Repeat for another pair of parallel chords. The two lines interesect at F.

15th Putnam 1955

© John Scholes

jscholes@kalva.demon.co.uk

26 Nov 1999