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 2ndigit 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^{i1} 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!
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