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Finding points and loci for a given square...

Source: Vietnam MO 1984 P3

March 20, 2011

geometry unsolvedgeometry

Problem Statement

A square

ABCD

of side length

2a

is given on a plane

\Pi

. Let

S

be a point on the ray

Ax

perpendicular to

\Pi

such that

AS = 2a.

(a)

Let

M \in BC

and

N \in CD

be two variable points.

i

. Find the positions of

M,N

such that

BM + DN \ge \frac{3}{2}

, planes

SAM

and

SMN

are perpendicular and

BM \cdot DN

is minimum.

ii

. Find

M

and

N

such that

\angle MAN = 45^{\circ}

and the volume of

SAMN

attains an extremum value. Find these values.

(b)

Let

Q

be a point such that

\angle AQB = \angle AQD = 90^{\circ}

. The line

DQ

intersects the plane

\pi

through

AB

perpendicular to

\Pi

at

Q'

.

i

. Find the locus of

Q'

.

ii

. Let

K

be the locus of points

Q

and let

CQ

meet

K

again at

R

. Let

DR

meets

\Pi

at

R'

. Prove that

sin^2 \angle Q'DB + sin^2 \angle R'DB

is independent of

Q

.

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