Escaping elevens

From a Romanian Selection Test:

What is the greatest possible length of a sequence of consecutive positive integers, so that none of the integers have a digit sum divisible by 11?



Children and Chocolate

Adapted from a Maths Battle in London:

Several children sit around a circular table, on which lies a collection of 100 chocolates. They proceed to take chocolates, after which it turns out that each child has taken either 6 fewer or 3 times as many chocolates as the child to their left. Prove that some chocolate was left on the table.


Five People

Here is a problem that appeared in a Maths Battle in London a couple of weeks ago:

Donald, Jack, Peter, Richard and Steven have, in some order, the surnames Donaldson, Jackson, Peterson, Richardson and Stevenson. Donald is 1 year older than Donaldson, Jack is 2 years older than Jackson, Peter is 3 years older than Peterson, and Richard is 4 years older than Richardson. Who out of Steven and Stevenson is older, and by how much?



From the Tournament of the Towns:

A balance and a set of metal weights are given, with no two the same. If any pair of these weights is placed in the left pan of the balance, then it is always possible to counterbalance them with one or several of the remaining weights placed in the right pan. What is the smallest possible number of weights in the set?



A stunner from the Baltic Way:

A family wears clothes of three colours: red, blue and green, with a separate identical laundry bin for each colour. At the beginning of the first week, all the bins are empty. Each week, the family generates 10kg of laundry (the proportion of each colour might vary week to week). The laundry is sorted by colour and placed in the bins. Next, the heaviest bin (only one if there are several) is emptied and its contents washed.

What is the minimum requirement (in kg) of each bin?