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Found: 2

Problems Number Theory Numeral systems Binary number system Algebra and arithmetic Algebraic geometry Cryptography Discrete Mathematics Algorithm Theory Theory of algorithms (other)

13–15 2.0

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.

Problem #PRU-66202

Problems Number Theory Numeral systems Binary number system Methods Algebraic methods Iterations Algebra Algebraic inequalities and systems of inequalities Number inequalities. Number comparison Calculus Real numbers Rational and irrational numbers

15–16 5.0

Author: A.K. Tolpygo

An irrational number $α$ , where $0 < α < \frac{1}{2}$ , is given. It defines a new number $α_{1}$ as the smaller of the two numbers $2 α$ and $1 - 2 α$ . 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 $n$ ?