2

Part of 1990 Poland - Second Round

Problems(1)

can this set of points be a tetrahedron ?

Source: Polish MO Recond Round 1990 p2

In space, a point

O

and a finite set of vectors

\overrightarrow{v_1},\ldots,\overrightarrow{v_n}

are given . We consider the set of points

P

for which the vector

\overrightarrow{OP}

can be represented as a sum

a_1 \overrightarrow{v_1} + \ldots + a_n\overrightarrow{v_n}

with coefficients satisfying the inequalities

0 \leq a_i \leq 1

( i = 1, 2, \ldots, n

). Decide whether this set can be a tetrahedron.

geometry3D geometrytetrahedron