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Found: 3

Problem #PRU-116021

Problems Algebra Algebraic inequalities and systems of inequalities Number inequalities. Number comparison

Find the largest natural number $n$ which satisfies $n^{200} < 5^{300}$ .

Problem #PRU-35052

Problems Algebra Algebraic inequalities and systems of inequalities Classical inequalities Probability and statistics Probability theory Discrete distribution Number inequalities. Number comparison

There are 30 students in the class. Prove that the probability that some two students have the same birthday is more than 50%.

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

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 $n$ ?