Our readers propose

Mr. Roman Makaj from Lublin, Poland, has observed that a certain extremal property of a circumference can be used to obtain a figure of minimal area. The property in question is the fact that among all curves of given length the circumference encloses the biggest area (on the plane). Mr. Makaj inscribes a square into the circumference and then takes the symmetric reflection of each arc determined by a side of the square with respect to that side. Thus he arrives at the conclusion that among curves of length

passing through all the vertices of the square with side a, the curve composed of the reflections of the arcs of the circumscribed circle (see the figure below)

encloses the least area.

We hope the reader will provide his own proof of the fact.

Consider the following problem: From a given number with an even number of digits subtract the number obtained by reversing the order of the digits in the former. Now, to this difference add the number obtained again by reversing the order of its digits. (if the difference had less digits than the initial number, add '0' at the beginning and then reverse). It turns out that if the initial number had 2n digits, in 45^n cases the result of the entire procedure will be the 2n-digit number composed of all 9's. Such an outcome is ensured by the following condition satisfied by the initial number (where a(i) is the digit standing by 10^{i-1} in the decimal expansion of the number):

In the remaining cases, when the above condition fails, the procedure yields a number with none of the digits 2, 3, 4, 5, 6, 7 in its decimal notation.

We have received this result with the corresponding proof from Mr.Tadeusz Boncler; we consider that the publication of the proof would deprive our reader of the pleasure of providing it by himself. Good luck!