MathDB

old vietanmese locus in space

Source: Vietnamese MO (VMO) 1964

August 22, 2018

geometry3-Dimensional GeometryLocus

Problem Statement

Let

P

be a plane and two points

A \in (P),O \notin (P)

. For each line in

(P)

through

A

, let

H

be the foot of the perpendicular from

O

to the line. Find the locus

(c)

of

H

. Denote by

(C)

the oblique cone with peak

(C)

and base

(c)

. Prove that all planes, either parallel to

\beta

or perpendicular to

OA

, intersect

(C)

by circles. Consider the two symmetric faces of

(C)

that intersect

(C)

by the angles

\alpha

and

\beta

respectively. Find a relation between

\alpha

and

\beta

.

Back to Problems View on AoPS