KVPY 2019 SB SX Exam Previous Papers

Let ${\mathit{\sigma }}_1,{\mathit{\sigma }}_2,{\mathit{\sigma }}_3$ be planes passing through the origin. Assume that ${\mathit{\sigma }}_1$ is perpendicular to the vector (1, 1, 1), ${\mathit{\sigma }}_2$ is perpendicular to a vector (a, b, c), and ${\mathit{\sigma }}_3$ is perpendicular to the vector $\left({\mathit{a}}^2,{\mathit{b}}^2,{\mathit{c}}^2\right)$. What are all the positive values of a, b, and c so that ${\mathit{\sigma }}_1\cap {\mathit{\sigma }}_2\cap {\mathit{\sigma }}_3$ is a single point ?