Problems

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?

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?

To transmit messages by telegraph, each letter of the Russian alphabet () ( and 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 is represented in the form 00000, letter -00001, letter -10111, letter -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 is received. Find the word you sent.

Author: A.K. Tolpygo

An irrational number $\alpha$, where $0<\alpha <\frac{1}{2}$, is given. It defines a new number ${\alpha }_1$ as the smaller of the two numbers $2\alpha$ and $1-2\alpha$. For this number, ${\alpha }_2$ is determined similarly, and so on.

a) Prove that for some $n$ the inequality ${\alpha }_n<3/16$ holds.

b) Can it be that ${\alpha }_n>7/40$ for all positive integers $n$?

When we write $137$ in decimal, we mean $1\times {10}^2+3\times 10+7\times 1$. If we write it instead using powers of $2$, we have $137=1\times 2^7+0\times 2^6+0\times 2^5+0\times 2^4+1\times 2^3+0\times 2^2+0\times 2^1+1\times 2^0$. To tell apart binary representation from decimals, we can use the following notation: $137=(10001001{)}_2$.

What is the number $273$ in binary? Note that using binary is useful for finding whether a particular Nim game is a winning position or a losing position.