Question 33
Suppose $a, b, c$ are non-zero real or complex numbers, and $x, y, z$ satisfy the following equations.
$$
bz + cy = a, \ cx + az = b, \ bx + ay = c.
$$
Prove that
$$
\frac{3}{4} \le |x|^2 + |y|^2 + |z|^2,
$$
and give an example to show that this inequality is best possible.
Deduce, or prove otherwise, that in any triangle $ABC$
$$
\cos^2 A + \cos^2 B + \cos^2 C \ge \frac{3}{4}.
$$
$$
bz + cy = a, \ cx + az = b, \ bx + ay = c.
$$
Prove that
$$
\frac{3}{4} \le |x|^2 + |y|^2 + |z|^2,
$$
and give an example to show that this inequality is best possible.
Deduce, or prove otherwise, that in any triangle $ABC$
$$
\cos^2 A + \cos^2 B + \cos^2 C \ge \frac{3}{4}.
$$