There are one hundred natural numbers, they are all different, and sum up to 5050. Can you find those numbers? Are they unique, or is there another bunch of such numbers?
Can \(100\) weights of masses \(1,2,3,\dots,99,100\) be arranged into \(10\) piles, all of different total masses, so that the heavier a pile is, the fewer weights it contains?
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 a volleyball tournament teams play each other once. A win gives the team 1 point, a loss 0 points. It is known that at one point in the tournament all of the teams had different numbers of points. How many points did the team in second last place have at the end of the tournament, and what was the result of its match against the eventually winning team?
Is the sum of the numbers \(1 + 2 + 3 + \dots + 1999\) divisible by 1999?