Author: A.K. Tolpygo
An irrational number α, where 0 $<$α $<$½, is given. It defines a new number $α_1$ as the smaller of the two numbers 2α and 1 - 2α. For this number, $α_2$ is determined similarly, and so on.
a) Prove that for some n the inequality $α_n <3/16$ holds.
b) Can it be that $α_n> 7/40$ for all positive integers n?
To transmit messages by telegraph, each letter of the Russian alphabet $($а, б, в, г, д, е, ё, ж, з, и, й, к, л, м, н, о, п, р, с, т, у, ф, х, ц, ч, ш, щ, ъ, ы, ь, э, ю, я$)$ $($E and Ё are counted as identical$)$ is represented as a five-digit combination of zeros and ones corresponding to the binary number of the given letter in the alphabet $($letter numbering starts from zero$)$. For example, the letter A is represented in the form 00000, letter B-00001, letter Ч-10111, letter Я-11111. Transmission of the five-digit combination is made via a cable containing five wires. Each bit is transmitted on a separate wire. When you receive a message, Cryptos has confused the wires, so instead of the transmitted word, a set of letters ЭАВЩОЩИ is received. Find the word you sent.
Suppose you have 127 1p coins. How can you distribute them among 7 coin pouches such that you can give out any amount from 1p to 127p without opening the coin pouches?
Michael thinks of a number no less than 1 and no greater than 1000. Victoria is only allowed to ask questions to which Michael can answer “yes” or “no” (Michael always tells the truth). Can Victoria figure out which number Michael thought of by asking 10 questions?