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 1997/98

Number theory

  1. Is it possible to pair integers 1, 2, ..., 50, if we require that all sums of pairs should be different primes?

  2. Prove that the number 1998 is a sum of several natural numbers a1, a2, ..., an such that

    1/a1 + 1/a2 + ... + 1/an = 1.

    if we know that numbers 33, 34, ..., 73 have the same property.

  3. Let integers x, y are prime to each other and xy>1. Prove that x + y is not a divisor of xn + yn if the exponent n is an even integer.

  4. We can obtain the number A=100010011002...99989999 by joining the digits of all 4-digit integers.
    Prove, that if A=mn and m,n are integers, then n=1.

  5. Prove, that we can find infinite number of integers 5n (n is a positive integer, too) last 1998 digits of which have different parity than their two neighbouring digits.