Question 84
Given two finite sets $A$ and $B$ of pairs of real numbers. For any three pairs $(a_1, b_1)$, $(a_2, b_2)$ and $(a_3, b_3)$ from $A$ there exists a pair $(c, d)$ from $B$ such that
$$
a_1c + b_1d \ge 0, \quad a_2c + b_2d \ge 0 \quad \text{and} \quad a_3c + b_3d \ge 0.
$$
Prove that the $B$ contains a pair $(\gamma, \delta)$ such that $\alpha\gamma + \beta\delta \ge 0$ for each pair $(\alpha, \beta)$ from $A$.
$$
a_1c + b_1d \ge 0, \quad a_2c + b_2d \ge 0 \quad \text{and} \quad a_3c + b_3d \ge 0.
$$
Prove that the $B$ contains a pair $(\gamma, \delta)$ such that $\alpha\gamma + \beta\delta \ge 0$ for each pair $(\alpha, \beta)$ from $A$.