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?
How many integers are there from 0 to 999999, in the decimal notation of which there are no two identical numbers next to each other?
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?
On an infinitely long strip of paper, we write an endless row of digits.
We start by writing \(1,2,3,4\). After that, each new digit is chosen like this: add the previous four digits and write down only the last digit of that sum.
So the beginning looks like \(1234096\dots\).
Will the four digits \(8123\) ever appear next to each other somewhere in this endless row?
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?
In a row there are 2023 numbers. The first number is 1. It is known that each number, except the first and the last, is equal to the sum of two neighboring ones. Find the last number.
In good conditions, bacteria in a Petri cup spread quite fast, doubling every second. If there was initially one bacterium, then in \(32\) seconds the bacteria will cover the whole surface of the cup.
Now suppose that there are initially \(4\) bacteria. At what time will the bacteria cover the surface of the cup?
Among the first \(20\) Fibonacci numbers: \(F_0 = 0,F_1 = 1,F_2 = 1, F_3 = 2, F_4 = 3,..., F_{20} = 6765\) find all numbers whose digit-sum is equal to their index. For example, \(F_1=1\) fits the description, but \(F_{20} = 6765\) does not, since \(6+7+6+5 \neq 20\).