p1. Cities A, B, C, D and E are located next to each other along the highway at a distance of 5 km from each other. The bus runs along the highway from city A to city E and back. The bus consumes 20 liters of gasoline for every 100 kilometers. In which city will a bus run out of gas if it initially had 150 liters of gasoline in its tank?
p2. Find the minimum four-digit number whose product of all digits is 729. Explain your answer.
p3. At the parade, soldiers are lined up in two lines of equal length, and in the first line the distance between adjacent soldiers is 20% greater than in the second (there is the same distance between adjacent soldiers in the same line). How many soldiers are in the first rank if there are 85 soldiers in the second rank?
p4. It is known about three numbers that the sum of any two of them is not less than twice the third number, and the sum of all three is equal to 300. Find all triplets of such (not necessarily integer) numbers.
p5. The tourist fills two tanks of water using two hoses. 2.9 liters of water flow out per minute from the first hose, 8.7 liters from the second. At that moment, when the smaller tank was half full, the tourist swapped the hoses, after which both tanks filled at the same time. What is the capacity of the larger tank if the capacity of the smaller one is 12.5 liters?
p6. Is it possible to mark 6 points on a plane and connect them with non-intersecting segments (with ends at these points) so that exactly four segments come out of each point?
p7. Petya wrote all the natural numbers from 1 to 1000 and circled those that are represented as the difference of the squares of two integers. Among the circled numbers, which numbers are more even or odd?
p8. On a sheet of checkered paper, draw a circle of maximum radius that intersects the grid lines only at the nodes. Explain your answer.
p9. Along the railway there are kilometer posts at a distance of 1 km from each other. One of them was painted yellow and six were painted red. The sum of the distances from the yellow pillar to all the red ones is 14 km. What is the maximum distance between the red pillars?
p10. The island nation is located on 100 islands connected by bridges, with some islands also connected to the mainland by a bridge. It is known that from each island you can travel to each (possibly through other islands). In order to improve traffic safety, one-way traffic was introduced on all bridges. It turned out that from each island you can leave only one bridge and that from at least one of the islands you can go to the mainland. Prove that from each island you can get to the mainland, and along a single route.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here. algebrageometrycombinatoricsnumber theorySoros Olympiad