## Task 1 - 280831

Two travel-loving friends $A$ and $B$ decide to walk towards each other on a hiking trail of $30 \mathrm{~km}$. At the beginning, $A$ is at one endpoint of the trail, and $B$ is at the other endpoint. They coordinate their plan over the phone and assume that each of them will maintain their personal walking speed throughout the entire journey. This leads to the following statements:

(1) If $A$ starts 2 hours earlier than $B$, they will meet $2 \frac{1}{2}$ hours after $B$ starts.

(2) If, however, $B$ starts 2 hours earlier than $A$, they will meet 3 hours after $A$ starts.

Show that under these assumptions, if statements (1) and (2) are true, the walking speeds of $A$ and $B$ are uniquely determined; determine these speeds!

Verify that the reverse is also true: If $A$ and $B$ have the determined speeds, then statements (1) and (2) are true.
a) If statements (1) and (2) are true, then the following applies to the values \( v_{A}, v_{B} \) of the speeds of A and B, measured in \(\frac{\mathrm{km}}{\mathrm{h}}\):

If A and B each travel from their starting point to the meeting point for \(4 \frac{1}{2}\) hours and \(2 \frac{1}{2}\) hours respectively, according to (1), they cover distances whose lengths in \(\mathrm{km}\) are \(4 \frac{1}{2} \cdot v_{A}\) and \(2 \frac{1}{2} \cdot v_{B}\) respectively. This leads to

\[
4 \frac{1}{2} \cdot v_{A} + 2 \frac{1}{2} \cdot v_{B} = 30
\]

If A and B each travel from their starting point to the meeting point for 3 hours and 5 hours respectively, according to (2), it follows correspondingly that

\[
3 \cdot v_{A} + 5 \cdot v_{B} = 30
\]

From (3) it follows that \(9 \cdot v_{A} + 5 \cdot v_{B} = 60\); from this and (4) it follows that \(6 \cdot v_{A} = 30, v_{A} = 5\). Substituting this into (4) gives \(5 \cdot v_{B} = 15, v_{B} = 3\).

Thus, the marching speeds, if (1) and (2) are true, are uniquely determined; they are \(5 \frac{\mathrm{km}}{\mathrm{h}}\) and \(3 \frac{\mathrm{km}}{\mathrm{h}}\) respectively.

II. If A and B have these speeds, then the following follows:

In \(4 \frac{1}{2}\) hours, A covers the partial distance of \(22 \frac{1}{2} \mathrm{~km}\) and B in \(2 \frac{1}{2}\) hours the partial distance of \(7 \frac{1}{2} \mathrm{~km}\). Since \(22 \frac{1}{2} + 7 \frac{1}{2} = 30\), they meet, so (1) is true.

In 3 hours, A covers the partial distance of \(15 \mathrm{~km}\) and B in 5 hours the partial distance of \(15 \mathrm{~km}\). Since \(15 + 15 = 30\), they meet, so (2) is also true.