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Found: 3

Problems Methods Algebraic methods Iterations Mathematical induction Mathematical induction (other) Processes and operations Calculus Real numbers Rational and irrational numbers Discrete Mathematics Algorithm Theory Theory of algorithms (other)

14–16 4.0

With a non-zero number, the following operations are allowed: $x \to \frac{1 + x}{x}$ , $x \to \frac{1 - x}{x}$ . Is it true that from every non-zero rational number one can obtain each rational number with the help of a finite number of such operations?

Problem #PRU-35476

Problems Calculus Functions of one variable. Continuity Methods Algebraic methods Iterations

15–16 3.0

Let $f (x)$ be a polynomial about which it is known that the equation $f (x) = x$ has no roots. Prove that then the equation $f (f (x)) = x$ does not have any roots.

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 $n$ ?