Problems

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 $[0,1]$.

Is it possible to:

a) load two coins so that the probability of “heads” and “tails” were different, and the probability of getting any of the combinations “tails, tails,” “heads, tails”, “heads, heads” be the same?

b) load two dice so that the probability of getting any amount from 2 to 12 would be the same?

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 point $O$, lying inside the triangle $ABC$, is connected by segments with the vertices of the triangle. Prove that the variance of the set of angles $AOB$, $AOC$ and $BOC$ is less than a) $10{\pi }^2/27$; b) $2{\pi }^2/9$.

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.