A professional tennis player plays at least one match each day for training purposes. However in order to ensure he does not over-exert himself he plays no more than 12 matches a week. Prove that it is possible to find a group of consecutive days during which the player plays a total of 20 matches.
100 fare evaders want to take a train, consisting of 12 coaches, from the first to the 76th station. They know that at the first station two ticket inspectors will board two coaches. After the 4th station, in the time between each station, one of the ticket inspectors will cross to a neighbouring coach. The ticket inspectors take turns to do this. A fare evader can see a ticket inspector only if the ticket inspector is in the next coach or the next but one coach. At each station each fare evader has time to run along the platform the length of no more than three coaches – for example at a station a fare evader in the 7th coach can run to any coach between the 4th and 10th inclusive and board it. What is the largest number of fare evaders that can travel their entire journey without ever ending up in the same coach as one of the ticket inspectors, no matter how the ticket inspectors choose to move? The fare evaders have no information about the ticket inspectors beyond that which is given here, and they agree their strategy before boarding.
A road of length 1 km is lit with streetlights. Each streetlight illuminates a stretch of road of length 1 m. What is the maximum number of streetlights that there could be along the road, if it is known that when any single streetlight is extinguished the street will no longer be fully illuminated?
In the number \(1234096\dots\) each digit, starting with the 5th digit, is equal to the final digit of the sum of the previous 4 digits. Will the digits 8123 ever occur in a row in this number?
7 natural numbers are written around the edges of a circle. It is known that in each pair of adjacent numbers one is divisible by the other. Prove that there will be another pair of numbers that are not adjacent that share this property.
Is it possible to transport 50 stone blocks, whose masses are equal to \(370, 372,\dots, 468\) kg, from a quarry on seven 3-tonne trucks?
Is it possible to arrange 1000 line segments in a plane so that both ends of each line segment rest strictly inside another line segment?
Some open sectors – that is sectors of circles with infinite radii – completely cover a plane. Prove that the sum of the angles of these sectors is no less than \(360^\circ\).
On a line, there are 50 segments. Prove that either it is possible to find some 8 segments all of which have a shared intersection, or there can be found 8 segments, no two of which intersect.
10 magazines lie on a coffee table, completely covering it. Prove that you can remove five of them so that the remaining magazines will cover at least half of the table.