[ Pseudoscalar product ]

Three runners $A, B$ and $C$ are running on parallel tracks at constant speeds. At the initial moment, the area of triangle $A B C$ is 2, and after 5 seconds, it is 3. What could it be after another 5 seconds?

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Let at the initial moment, i.e., at $t=0$, $\overrightarrow{A B}=\mathbf{v}$ and $\overrightarrow{A C}=\mathbf{w}$. Then at time $t$, we get $\overrightarrow{A B}=\mathbf{v}+t(\mathbf{b}-\mathbf{a})$ and $\overrightarrow{A C}=\mathbf{w}+t(\mathbf{c}-\mathbf{a})$, where $\mathbf{a}, \mathbf{b}$, and $\mathbf{c}$ are the velocity vectors of runners $A, B$, and $C$. Since vectors $\mathbf{a}, \mathbf{b}$, and $\mathbf{c}$ are parallel, then $(\mathbf{b}-\mathbf{a}) \vee (\mathbf{c}-\mathbf{a})=0$, which means $|S(A, B, C)|=|\overrightarrow{A B} \vee \overrightarrow{A C}| / 2=|x+y t|$, where $x$ and $y$ are some constant numbers. Solving the system $|x|=2,|x+5 y|=3$, we get two solutions, giving for the dependence of the area of triangle $A B C$ on time $t$ the expressions $|2+(t / 5)|$ and $|2 - t|$. Therefore, at $t=10$, the area can take the values 4 and 8.

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