Problems

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 $[0,1]$.

On a calculator keypad, there are the numbers from 0 to 9 and signs of two actions (see the figure). First, the display shows the number 0. You can press any keys. The calculator performs the actions in the sequence of clicks. If the action sign is pressed several times, the calculator will only remember the last click.

a) The button with the multiplier sign breaks and does not work. The Scattered Scientist pressed several buttons in a random sequence. Which result of the resulting sequence of actions is more likely: an even number or an odd number?

b) Solve the previous problem if the multiplication symbol button is repaired.

In a numerical set of $n$ numbers, one of the numbers is 0 and another is 1.

a) What is the smallest possible variance of such a set of numbers?

b) What should be the set of numbers for this?

The numbers $a_1,a_2,\dots ,a_{1985}$ are the numbers $1,2,\dots ,1985$ rearranged in some order. Each number $a_k$ is multiplied by its number $k$, and then the largest number is chosen among the resulting 1985 products. Prove that it is not less than ${993}^2$.

Prove that there is a number of the form

a) $1989\dots 19890\dots 0$ (the number 1989 is repeated several times, and then there are a few zeros), which is divisible by 1988;

b) $1988\dots 1988$, which is divisible by 1989.

Solve the inequality: $\lfloor x\rfloor \times \{x\}<x-1$.

On a plane, there are 1983 points and a circle of unit radius. Prove that there is a point on the circle, from which the sum of the distances to these points is no less than 1983.