Problems

At all rational points of the real line, integers are arranged. Prove that there is a segment such that the sum of the numbers at its ends does not exceed twice the number on its middle.

Prove that for any positive integer $n$ the inequality

is true.

The functions $f(x)-x$ and $f(x^2)-x^6$ are defined for all positive $x$ and increase. Prove that the function

also increases for all positive $x$.

Prove that if the numbers $x,y,z$ satisfy the following system of equations for some values of $p$ and $q$: $$\begin{array}{rl}y& =x^2+px+q,\\ z& =y^2+py+q,\\ x& =z^2+pz+q,\end{array}$$ then the inequality $x^{2}y+y^{2}z+z^{2}x\ge x^{2}z+y^{2}x+z^{2}y$ is satisfied.

Prove that in any infinite decimal fraction you can rearrange the numbers so that the resulting fraction becomes a rational number.

We are given rational positive numbers $p,q$ where $1/p+1/q=1$. Prove that for positive $a$ and $b$, the following inequality holds: $ab\le \frac{a^p}{p}+\frac{b^q}{q}$.

Let $p$ and $q$ be positive numbers where $1/p+1/q=1$. Prove that $$a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n\le (a_1^p+\dots a_n^p{)}^{1/p}(b_1^q+\cdots +b_n^q{)}^{1/q}$$ The values of the variables are considered positive.

Find the largest value of the expression $a+b+c+d-ab-bc-cd-da$, if each of the numbers $a$, $b$, $c$ and $d$ belongs to the interval $[0,1]$.

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 $n$?

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