Is there a sequence of natural numbers in which every natural number occurs exactly once, and for any \(k = 1, 2, 3, \dots\) the sum of the first \(k\) terms of the sequence is divisible by \(k\)?
Which term in the expansion \((1 + \sqrt 3)^{100}\) will be the largest by the Newton binomial formula?
Prove that in any infinite decimal fraction you can rearrange the numbers so that the resulting fraction becomes a rational number.
Author: I.I. Bogdanov
Peter wants to write down all of the possible sequences of 100 natural numbers, in each of which there is at least one 4 or 5, and any two neighbouring terms differ by no more than 2. How many sequences will he have to write out?