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

Problem

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 $n$ ?

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