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 spring series 1996/97

1997

  1. This year is a very special one for two brothers Kurt and Horst. Kurt's age is equal to the sum of digits of Horst's year of birth and Horst's age is equal to the sum of digitis of Kurt's year of birth. Both are born on the January 1st and Kurt is 8 years older. How old are they?

  2. Prove that there exist at least as many unsocial people as dependent people in the world. Somebody is unsocial if she or he has less than 1997 acquaintances and dependent if all her or his acquaintances are unsocial.

  3. Find all integers n such that 1999n is terminated by four digits: 1997. (7 is least significant and numbers are written in decimal).

  4. 1997 greenpeace cyclists want to ride from Bratislava to Moscow in order to protest against nuclear energy. This route is about 1997 kilometers long but they have only 1996 bicycles. A cyclist walks at speed of 5 kilometers per hour and rides bike at speed of 20 km/h. (suppose there are no bike-burglars on the way.)

  5. Let the area of polygon with 1997 vertices A1,...,A1997 is S. Let's construct a polygon B1...B1997 as follows: We double the length of the edge A1997A1 and mark the new end point behind A1 as B1 (i.e. |A1997A1|=|A1B1|). Points B2, B3, ... B1997 are constructed by prolonging edges A1A2, ..., A1996A1997 the same way. Prove that the area of polygon B1,...,B1997 is not greater then 5S.