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Let a, b, and c be real numbers. Assume there exist real numbers x, y, z, not all zero, satisfying x = cy + bz, y = az + cx, and z = bx + ay. Then the value of a² + b² + c² + 2abc is

1. 2
2. -1
3. 0
4. 1

Correct answer: 1

Solution

Writing the three relations as a homogeneous system in x,y,z, a non-trivial solution requires the coefficient determinant to vanish. Expanding |[-1,c,b],[c,-1,a],[b,a,-1]|=0 gives 1-(a^2+b^2+c^2)-2abc=0, so a^2+b^2+c^2+2abc=1.

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