Advanced book on Mathematics Olympiad

446 Algebra

Consider the pointsP(a 11 ,a 21 ,a 31 ), Q(a 12 ,a 22 ,a 32 ), R(a 13 ,a 23 ,a 33 )in three-di-
mensional Euclidean space. It is enough to find a point in the interior of the triangle
PQRwhose coordinates are all positive, all negative, or all zero.
LetP′,Q′,R′be the projections ofP , Q, Ronto thexy-plane. The hypothesis
implies thatP′,Q′,andR′lie in the fourth, second, and third quadrant, respectively.

Case1. The originOis in the exterior or on the boundary of the triangleP′Q′R′(Fig-
ure 63).

x

P

O

y

S

Q

R’

’

’

’

Figure 63

Denote byS′the intersection of the segmentsP′Q′andOR′, and letSbe the point
on the segmentPQwhose projection isS′. Note that thez-coordinate of the pointSis
negative, since thez-coordinates ofP′andQ′are negative. Thus any point in the interior
of the segmentSRsufficiently close toShas all coordinates negative, and we are done.

Case2. The originOis in the interior of the triangleP′Q′R′(Figure 64).

x

R

O

y

Q

P

’

’

’

Figure 64

LetTbe the point inside the trianglePQRwhose projection isO.IfT=O, we are
done. Otherwise, if thez-coordinate ofTis negative, choose a pointSclose to it inside