BRATISLAVA MATHEMATICS CORRESPONDENCE SEMINAR
Faculty of Mathematics and Physics of Comenius University
Union of Slovak Mathematicians and Physicists
Centre of Spare Time - IUVENTA
BMCS - Problem set of the 1st fall series
1997/98
Math caviar
-
A young cook has 18 liters of hot tea in her pot. She wants to put 6 liters
of the tea into each of the 2 cans with the volume of 7 liters for two
groups of her friends. She has only a big ladle with the volume of 4 liters.
Can you help her to divide the tea?
-
Red hood has written down n integers on the paper and decided to select
several of them (but at least one) so that the sum of the selected numbers will
be dividable by n. Prove that she will succeed regardless of which
numbers has she written on the paper.
-
Sherlock Holmes found a notice with several integers, all of them consisting of
n digits. After a more detailed investigation, he discovered that all
numbers contained only digits 5 and 8 and any two numbers differed at least
in 2 digits. What is the maximum number of the integers that could have been
written on the notice?
-
There are red and blue teeth on the necklace. The manufacturer allows to
perform the following changes:
- Insert a red tooth between any two teeth while changing the colors of
the two new neighbors.
- Remove one red tooth (if there are at least three teeth on the necklace)
and change the colors of its two original neighbors.
The basic model contains only two red teeth. Is it possible to perform several
changes so that the resultant will contain
a) one red and one blue tooth
b) one red tooth and one hundred blue teeth?
c) only two blue teeth
(notice: only the c) part is scored).
-
"All of a sudden, several lines appeared in the square 1x1. The sum of their
lengths is 18. All of them are parallel with one of the square's edges and
they divide the square into several parts. Mulder thinks that the area of
each part is smaller than 0.01. I'm afraid he is really wrong now." Is the
agent Scully right?