Santa Delivers Presents

We need to 1st find the Earth's Circumference in accord with the given assumptions. Approximating the value of pie as $22/7$ does not work because of the Earth's Large Diameter. ( I know because I got two tries one with pie as $3.14$ and one with pie as $22/7$ both wrong.) So we use the more accurate value of the Earth's Circumference as $π * 12742$ . The amount of great circles we have to traverse is given by $180/10$ which is $18$ . Hence our total distance is $π * 12742 * 18$ . To get speed simply divide it by time giving us $(π * 12742*18)/12$ . Just put these values in Wolfram Alpha to get the final answer which is minimum speed $\boxed{60,045.2}$ . Now enter last three digits after rounding up the value.

one trip around the world is $R\pi$ . Since he is traveling $\frac{180}{10}=18$ times around the world, because he doesn't want to travel the same path in the different direction. In total that is $18R\pi$ . The journey has to finish in 12h, so $v=\frac{18R\pi}{t}=\frac{18\times 12742Km\times \pi}{12h}=60045.26$ . He mustn't travel slower, and since we are asked the smallest possible integer average that is $60\boxed{046}$

Smallest possible integer average speed = Ceiling(Total Distance Traveled in km / Time taken in hours) = Ceiling(18 π 12742/10) = Ceiling(60045.2604) = 60046 km/h.