Two people $A$ and $B$ need to get from point $M$ to point $N$, which is 15 km away from $M$, as quickly as possible. On foot, they can travel at a speed of 6 km/h. In addition, they have a bicycle that can be ridden at a speed of 15 km/h. $A$ sets out on foot, while $B$ rides the bicycle until meeting pedestrian $C$, who is walking from $N$ to $M$. After the meeting, $B$ continues on foot, and $C$ rides the bicycle until meeting $A$ and hands over the bicycle to him, who then rides it to $N$. When should pedestrian $C$ leave from $N$ so that the time spent by $A$ and $B$ on the journey to $N$ is minimized? ($C$ walks at the same speed as $A$ and $B$; the travel time is calculated from the moment $A$ and $B$ leave $M$ until the last of them arrives in $N$.)
To ensure that $A$ and $B$ spend the least amount of time on the road, they should arrive at $N$ simultaneously, meaning they should walk the same distances. Indeed, if $A$ arrives at $N$ before $B$, $A$ can give part of his time to $B$: $A$ can get on the bicycle a little later, allowing $B$ to ride the bicycle a little longer. As a result, the "total" arrival time will decrease. Similarly, it is not advantageous for $B$ to arrive at $N$ before $A$. For further details, see problem 77940.

## Answer

$3 / 11$ hours before $A$ and $B$ leave from $M$.

A thousand points are the vertices of a convex thousand-sided polygon, inside which there are another five hundred points such that no three of the five hundred lie on the same line. The thousand-sided polygon is cut into triangles in such a way that all the specified 1500 points are vertices of the triangles, and these triangles have no other vertices. How many triangles will result from such a cutting?

## Solution

Answer: 1998. The sum of all the angles of the resulting triangles is equal to the sum of the angles of the thousand-sided polygon plus 500 angles of $360^{\circ}$, corresponding to the 500 internal points. Thus, the sum of the angles of the triangles is $998 \cdot 180^{\circ} + 500 \cdot 360^{\circ} = (998 + 2 \cdot 500) \cdot 180^{\circ}$. Therefore, the number of resulting triangles is $998 + 2 \cdot 500 = 1998$.