[b]p1.[/b] Ben and his dog are walking on a path around a lake. The path is a loop $500$ meters around. Suddenly the dog runs away with velocity $10$ km/hour. Ben runs after it with velocity $8$ km/hour. At the moment when the dog is $250$ meters ahead of him, Ben turns around and runs at the same speed in the opposite direction until he meets the dog. For how many minutes does Ben run?


[b]p2.[/b] The six interior angles in two triangles are measured. One triangle is obtuse (i.e. has an angle larger than $90^o$) and the other is acute (all angles less than $90^o$). Four angles measure $120^o$, $80^o$, $55^o$ and $10^o$. What is the measure of the smallest angle of the acute triangle?


[b]p3.[/b] The figure below shows a $ 10 \times 10$ square with small $2 \times 2$ squares removed from the corners. What is the area of the shaded region?
[img]https://cdn.artofproblemsolving.com/attachments/7/5/a829487cc5d937060e8965f6da3f4744ba5588.png[/img]


[b]p4.[/b] Two three-digit whole numbers are called relatives if they are not the same, but are written using the same triple of digits. For instance, $244$ and $424$ are relatives. What is the minimal number of relatives that a three-digit whole number can have if the sum of its digits is $10$?


[b]p5.[/b] Three girls, Ann, Kelly, and Kathy came to a birthday party. One of the girls wore a red dress, another wore a blue dress, and the last wore a white dress. When asked the next day, one girl said that Kelly wore a red dress, another said that Ann did not wear a red dress, the last said that Kathy did not wear a blue dress. One of the girls was truthful, while the other two lied. Which statement was true?


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
**p1.** Ben and his dog are walking on a path around a lake. The path is a loop  $500$  meters around. Suddenly the dog runs away with velocity  $10$  km/hour. Ben runs after it with velocity  $8$  km/hour. At the moment when the dog is  $250$  meters ahead of him, Ben turns around and runs at the same speed in the opposite direction until he meets the dog. For how many minutes does Ben run?

1. Convert the velocities from km/hour to meters/minute:
   \[
   10 \text{ km/hour} = 10 \times \frac{1000 \text{ meters}}{60 \text{ minutes}} = \frac{10000}{60} \text{ meters/minute} \approx 166.67 \text{ meters/minute}
   \]
   \[
   8 \text{ km/hour} = 8 \times \frac{1000 \text{ meters}}{60 \text{ minutes}} = \frac{8000}{60} \text{ meters/minute} \approx 133.33 \text{ meters/minute}
   \]

2. Let \( t \) be the time in minutes it takes for Ben and the dog to meet. The dog is initially 250 meters ahead of Ben. When Ben turns around, they are running towards each other with a combined speed of:
   \[
   166.67 + 133.33 = 300 \text{ meters/minute}
   \]

3. The distance between them is 250 meters. The time \( t \) it takes for them to meet is:
   \[
   t = \frac{250 \text{ meters}}{300 \text{ meters/minute}} = \frac{5}{6} \text{ minutes} \approx 0.833 \text{ minutes}
   \]

The final answer is \(\boxed{0.833 \text{ minutes}}\).