MathDB

Given real numbers

(a_i)

and

(b_i)

(for

i=1,2,3,4

) such that

a_1 b _2 \ne a_2 b_1 .

Consider the set of all solutions

(x_1 ,x_2 ,x_3 , x_4)

of the simultaneous equations

a_1 x_1 +a _2 x_2 +a_3 x_3 +a_4 x_4 =0 \;\; \text{and}\;\; b_1 x_1 +b_2 x_2 +b_3 x_3 +b_4 x_4 =0

for which no

x_i

is zero. Each such solution generates a

4

-tuple of plus and minus signs (by considering the sign of

x_i

).
[*] Determine, with proof, the maximum number of distinct

4

-tuples possible. [*] Investigate necessary and sufficient conditions on

(a_i)

and

(b_i)

such that the above maximum of distinct

4

-tuples is attained.

A6