Think and solve!

With this issue we start publishing problems traditionally offered to readers of the traditional Delta. Since 1974, when Delta made its first public appearance, over 800 mathematical problems and over 450 problems in physics have occupied our readers' minds, not counting hundreds of problems included in our continuous mathematical and physical leagues. In most cases their solution requires more thinking than computing, and finding an elegant solution can provide much satisfaction.

The problems are always published with a solution, hidden somewhere in the same issue of the journal. However, we will not provide solutions here. Try for yourself. And if you really cannot find anything and you are sure you do not want to continue trying, but you still wish to know the solution, write to or and ask for one. In order to make our communication easier, we preserve the original numbers of the problems as they appeared in Delta. You may quote them when asking for a solution. But before you ask, think and solve!

We start with three mathematical problems by Krzysztof Oleszkiewicz.

M 804. In the small lost-in-the prairies town of Milky Cow 100 cowboys visit 100 local milk saloons every evening. Each of the cowboys chooses the place to spend the night over a pint of milk at random and independently of his colleagues' choices. The town sheriff intends to ban milk products in Milky Cow under the pretext that, anyway, most of the night 3/4 of the saloons remain empty. Milk-loving reader, please prove that the sheriff is wrong!

M 805. As a result of acute milk-hunger, sheriff's brain has produced a new illusion. This time our guardian of the law claims that most cowboys drink alone, which aggravates their addiction. Is he right?

M 806. In order to bring down the demand for milk in Milky Cow the sheriff decided to introduce drastic new laws. Now the distance between two saloons must be greater than 250 m., whereas the distance between any saloon and the one and only local prison must be less than 1 km. Is there enough room in town for 100 legal saloons?