There are \(12\) light bulbs placed on top of the numbers on a clock face. Initially, only the bulb at \(12\) is on. We can choose any \(6\) consecutive bulbs and change their state simultaneously, that is, if any were on, we turn them off, if any were off, we turn them on. Can we perform this operation multiple times so that in the end only the bulb at \(11\) is on?
In a certain state, there are three types of citizens:
A fool considers everyone a fool and themselves smart;
A modest clever person knows truth about everyone’s intellectual abilities and consider themselves a fool;
A confident clever person knows about everyone intellectual abilities correctly and consider themselves smart.
There are \(200\) deputies in the High Government. The Prime Minister conducted an anonymous survey of High Government members, asking how many smart people are there in the High Government. After reading everyone’s response he could not find out the number of smart people. But then the only member who did not participate in the survey returned from the trip. They filled out a questionnaire about the entire Government including themselves and after reading it the Prime Minister understood everything. How many smart could there be in the High Government (including the traveller)?
Winnie the Pooh has five friends, each of whom has pots of honey in their house: Tigger has \(1\) pot, Piglet has \(2\), Owl has \(3\), Eeyore has \(4\), and Rabbit has \(5\). Winnie the Pooh comes to visit each friend in turn, eats one pot of honey and takes the other pots with him. He came into the last house carrying \(10\) pots of honey. Whose house could Pooh have visited last?
Each integer on the number line is coloured either white or black. The numbers \(2016\) and \(2017\) are coloured differently. Prove that there are three identically coloured integers which sum to zero.
There are \(100\) non-zero numbers written in a circle. Between every two adjacent numbers, their product was written, and the previous numbers were erased. It turned out that the number of positive numbers after the operation coincides with the amount of positive numbers before. What is the minimum number of positive numbers that could have been written initially?
Each integer on the number line is coloured either yellow or blue. Prove that there is a colour with the following property: For every natural number \(k\), there are infinitely many numbers of this colour divisible by \(k\).
A new customer comes to the hotel and wants a room. It happened today that all the rooms are occupied. What should you do?
Now imagine you got \(10\) new guests arriving to the completely full hotel. What should you do now?
The next day you have even harder situation: to the hotel, where all the rooms are occupied arrives a bus with infinitely many new customers. In the bus all the seats have numbers \(1,2,3...\) corresponding to all natural numbers. How to deal with this one?
Imagine you have \(2\) new guests arriving to the full hotel. How do you accommodate them?